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Solving Equations
What is an equation.
An equation says that two things are equal. It will have an equals sign "=" like this:
That equations says:
what is on the left (x − 2) equals what is on the right (4)
So an equation is like a statement " this equals that "
What is a Solution?
A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .
Example: x − 2 = 4
When we put 6 in place of x we get:
which is true
So x = 6 is a solution.
How about other values for x ?
- For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
- For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .
In this case x = 6 is the only solution.
You might like to practice solving some animated equations .
More Than One Solution
There can be more than one solution.
Example: (x−3)(x−2) = 0
When x is 3 we get:
(3−3)(3−2) = 0 × 1 = 0
And when x is 2 we get:
(2−3)(2−2) = (−1) × 0 = 0
which is also true
So the solutions are:
x = 3 , or x = 2
When we gather all solutions together it is called a Solution Set
The above solution set is: {2, 3}
Solutions Everywhere!
Some equations are true for all allowed values and are then called Identities
Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities
Let's try θ = 30°:
sin(−30°) = −0.5 and
−sin(30°) = −0.5
So it is true for θ = 30°
Let's try θ = 90°:
sin(−90°) = −1 and
−sin(90°) = −1
So it is also true for θ = 90°
Is it true for all values of θ ? Try some values for yourself!
How to Solve an Equation
There is no "one perfect way" to solve all equations.
A Useful Goal
But we often get success when our goal is to end up with:
x = something
In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.
Example: Solve 3x−6 = 9
Now we have x = something ,
and a short calculation reveals that x = 5
Like a Puzzle
In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.
Here are some things we can do:
- Add or Subtract the same value from both sides
- Clear out any fractions by Multiplying every term by the bottom parts
- Divide every term by the same nonzero value
- Combine Like Terms
- Expanding (the opposite of factoring) may also help
- Recognizing a pattern, such as the difference of squares
- Sometimes we can apply a function to both sides (e.g. square both sides)
Example: Solve √(x/2) = 3
And the more "tricks" and techniques you learn the better you will get.
Special Equations
There are special ways of solving some types of equations. Learn how to ...
- solve Quadratic Equations
- solve Radical Equations
- solve Equations with Sine, Cosine and Tangent
Check Your Solutions
You should always check that your "solution" really is a solution.
How To Check
Take the solution(s) and put them in the original equation to see if they really work.
Example: solve for x:
2x x − 3 + 3 = 6 x − 3 (x≠3)
We have said x≠3 to avoid a division by zero.
Let's multiply through by (x − 3) :
2x + 3(x−3) = 6
Bring the 6 to the left:
2x + 3(x−3) − 6 = 0
Expand and solve:
2x + 3x − 9 − 6 = 0
5x − 15 = 0
5(x − 3) = 0
Which can be solved by having x=3
Let us check x=3 using the original question:
2 × 3 3 − 3 + 3 = 6 3 − 3
Hang On: 3 − 3 = 0 That means dividing by Zero!
And anyway, we said at the top that x≠3 , so ...
x = 3 does not actually work, and so:
There is No Solution!
That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!
This gives us a moral lesson:
"Solving" only gives us possible solutions, they need to be checked!
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Equation Word Problems Worksheets
This compilation of a meticulously drafted equation word problems worksheets is designed to get students to write and solve a variety of one-step, two-step and multi-step equations that involve integers, fractions, and decimals. These worksheets are best suited for students in grade 6 through high school. Click on the 'Free' icons to sample our handouts.
One Step Equation Word Problem Worksheets
Read and solve this series of word problems that involve one-step equations. Apply basic operations to find the value of unknowns.
(15 Worksheets)
Two-Step Equation Word Problems: Integers
Interpret this set of word problems that require two-step operations to solve the equations. Each printable worksheet has five word problems ideal for 6th grade, 7th grade, and 8th grade students.
Two-Step Equation Word Problems: Fractions and Decimals
Read each word problem and set up the two-step equation. Solve the equation and find the solution. This selection of worksheets includes both fractions and decimals.
MCQ - Two-Step Equation Word Problems
Pick the correct two-step equation that best matches word problems presented here. Evaluate the ability of students to solve two-step equations with this array of MCQ worksheets.
Multi-Step Equation Word Problems: Integers
Read each multi-step word problem in these high school pdf worksheets and set up the equation. Solve and find the value of the unknown. More than two steps are required to solve the problems.
Multi-step equation Word Problems: Fractions and Decimals
Write multi-step equations that involve both fractions and decimals based on the word problems provided here. Validate your responses with our answer keys.
Related Worksheets
» One-step Equation
» Two-step Equation
» Multi-step Equation
» Algebraic Identities
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Section 2.2 : Linear Equations
Solve each of the following equations and check your answer.
- \(4x - 7\left( {2 - x} \right) = 3x + 2\) Solution
- \(2\left( {w + 3} \right) - 10 = 6\left( {32 - 3w} \right)\) Solution
- \(\displaystyle \frac{{4 - 2z}}{3} = \frac{3}{4} - \frac{{5z}}{6}\) Solution
- \(\displaystyle \frac{{4t}}{{{t^2} - 25}} = \frac{1}{{5 - t}}\) Solution
- \(\displaystyle \frac{{3y + 4}}{{y - 1}} = 2 + \frac{7}{{y - 1}}\) Solution
- \(\displaystyle \frac{{5x}}{{3x - 3}} + \frac{6}{{x + 2}} = \frac{5}{3}\) Solution
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Quadratic Equations: Very Difficult Problems with Solutions
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Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to formulate queries.
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About solving equations
A value is said to be a root of a polynomial if ..
The largest exponent of appearing in is called the degree of . If has degree , then it is well known that there are roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, . This polynomial is considered to have two roots, both equal to 3.
One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.
Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.
How Wolfram|Alpha solves equations
For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. In some cases, linear algebra methods such as Gaussian elimination are used, with optimizations to increase speed and reliability. Other operations rely on theorems and algorithms from number theory, abstract algebra and other advanced fields to compute results. These methods are carefully designed and chosen to enable Wolfram|Alpha to solve the greatest variety of problems while also minimizing computation time.
Although such methods are useful for direct solutions, it is also important for the system to understand how a human would solve the same problem. As a result, Wolfram|Alpha also has separate algorithms to show algebraic operations step by step using classic techniques that are easy for humans to recognize and follow. This includes elimination, substitution, the quadratic formula, Cramer's rule and many more.
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The equations section of QuickMath allows you to solve and plot virtually any equation or system of equations. In most cases, you can find exact solutions to your equations. Even when this is not possible, QuickMath may be able to give you approximate solutions to almost any level of accuracy you require. It also contains a number of special commands for dealing with quadratic equations.
The Solve command can be uses to solve either a single equation for a single unknown from the basic solve page or to simultaneously solve a system of many equations in many unknowns from the advanced solve page . The advanced command allows you to specify whether you want approximate numerical answers as well as exact ones, and how many digits of accuracy (up to 16) you require. It also allows you to eliminate certain variables from the equations.
Go to the Solve page
The Plot command, from the Graphs section, will plot any function of two variables. In order to plot a single function of x, go to the basic equation plotting page , where you can enter the equation and specify the upper and lower limits on x that you want the graph to be plotted for. The advanced plotting page allows you to plot up to 6 equations on the one graph, each with their own color. It also gives you control over such things as whether or not to show the axes, where the axes should be located, what the aspect ratio of the plot should be and what the range of the dependent variable should be. All equations can be given in the explicit y = f(x) form or the implicit g(x,y) = c form.
Go to the Equation Plotting page
The Quadratics page contains 13 separate commands for dealing with the most common questions concerning quadratics. It allows you to : factor a quadratic function (by two different methods); solve a quadratic equation by factoring the quadratic, using the quadratic formula or by completing the square; rewrite a quadratic function in a different form by completing the square; calculate the concavity, x-intercepts, y-intercept, axis of symmetry and vertex of a parabola; plot a parabola; calculate the discriminant of a quadratic equation and use the discriminant to find the number of roots of a quadratic equation. Each command generates a complete and detailed custom-made explanation of all the steps needed to solve the problem.
Go to the Quadratics page
Introduction to Equations
By an equation we mean a mathematical sentence that states that two algebraic expressions are equal. For example, a (b + c) =ab + ac, ab = ba, and x 2 -1 = (x-1)(x+1) are all equations that we have been using. We recall that we defined a variable as a letter that may be replaced by numbers out of a given set, during a given discussion. This specified set of numbers is sometimes called the replacement set. In this chapter we will deal with equations involving variables where the replacement set, unless otherwise specified, is the set of all real numbers for which all the expressions in the equation are defined.
If an equation is true after the variable has been replaced by a specific number, then the number is called a solution of the equation and is said to satisfy it. Obviously, every solution is a member of the replacement set. The real number 3 is a solution of the equation 2x-1 = x+2, since 2*3-1=3+2. while 1 is a solution of the equation (x-1)(x+2) = 0. The set of all solutions of an equation is called the solution set of the equation.
In the first equation above {3} is the solution set, while in the second example {-2,1} is the solution set. We can verify by substitution that each of these numbers is a solution of its respective equation, and we will see later that these are the only solutions.
A conditional equation is an equation that is satisfied by some numbers from its replacement set and not satisfied by others. An identity is an equation that is satisfied by all numbers from its replacement set.
Example 1 Consider the equation 2x-1 = x+2
The replacement set here is the set of all real numbers. The equation is conditional since, for example, 1 is a member of the replacement set but not of the solution set.
Example 2 Consider the equation (x-1)(x+1) =x 2 -1 The replacement set is the set of all real numbers. From our laws of real numbers if a is any real number, then (a-1)(a+1) = a 2 -1 Therefore, every member of the replacement set is also a member of the solution set. Consequently this equation is an identity.
The replacement set for this equation is the set of real numbers except 0, since 1/x and (1- x)/x are not defined for x = 0. If a is any real number in the replacement set, then
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Praxis Core Math
Unit 1: lesson 4.
- Algebraic properties | Lesson
- Algebraic properties | Worked example
- Solution procedures | Lesson
- Solution procedures | Worked example
- Equivalent expressions | Lesson
- Equivalent expressions | Worked example
- Creating expressions and equations | Lesson
- Creating expressions and equations | Worked example
Algebraic word problems | Lesson
- Algebraic word problems | Worked example
- Linear equations | Lesson
- Linear equations | Worked example
- Quadratic equations | Lesson
- Quadratic equations | Worked example
What are algebraic word problems?
What skills are needed.
- Translating sentences to equations
- Solving linear equations with one variable
- Evaluating algebraic expressions
- Solving problems using Venn diagrams
How do we solve algebraic word problems?
- Define a variable.
- Write an equation using the variable.
- Solve the equation.
- If the variable is not the answer to the word problem, use the variable to calculate the answer.
What's a Venn diagram?
- Your answer should be
- an integer, like 6 6 6 6
- a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5
- a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4
- a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4
- an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75
- a multiple of pi, like 12 pi 12\ \text{pi} 1 2 pi 12, space, start text, p, i, end text or 2 / 3 pi 2/3\ \text{pi} 2 / 3 pi 2, slash, 3, space, start text, p, i, end text
- (Choice A) $ 4 \$4 $ 4 dollar sign, 4 A $ 4 \$4 $ 4 dollar sign, 4
- (Choice B) $ 5 \$5 $ 5 dollar sign, 5 B $ 5 \$5 $ 5 dollar sign, 5
- (Choice C) $ 9 \$9 $ 9 dollar sign, 9 C $ 9 \$9 $ 9 dollar sign, 9
- (Choice D) $ 14 \$14 $ 1 4 dollar sign, 14 D $ 14 \$14 $ 1 4 dollar sign, 14
- (Choice E) $ 20 \$20 $ 2 0 dollar sign, 20 E $ 20 \$20 $ 2 0 dollar sign, 20
- (Choice A) 10 10 1 0 10 A 10 10 1 0 10
- (Choice B) 12 12 1 2 12 B 12 12 1 2 12
- (Choice C) 24 24 2 4 24 C 24 24 2 4 24
- (Choice D) 30 30 3 0 30 D 30 30 3 0 30
- (Choice E) 32 32 3 2 32 E 32 32 3 2 32
- (Choice A) 4 4 4 4 A 4 4 4 4
- (Choice B) 10 10 1 0 10 B 10 10 1 0 10
- (Choice C) 14 14 1 4 14 C 14 14 1 4 14
- (Choice D) 18 18 1 8 18 D 18 18 1 8 18
- (Choice E) 22 22 2 2 22 E 22 22 2 2 22
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To obtain the solution for a word problem involving fractions, follow these steps:
1. State the problem clearly. 2. Add the phrase “ show me the step-by-step solution ” to your question.
Simplify fractions without
To simplify fractions without showing the steps, utilize the phrase “solve this fraction” before posing your inquiries.
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To simplify fractions showing the steps, utilize the phrase “solve this fraction with step by step solution” before posing your inquiries.
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Calculate equations, inequatlities, line equation and system of equations step-by-step.
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- x+2y=2x-5,\:x-y=3
Frequently Asked Questions (FAQ)
How do you solve algebraic expressions.
- To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.
What are the basics of algebra?
- The basics of algebra are the commutative, associative, and distributive laws.
What are the 3 rules of algebra?
- The basic rules of algebra are the commutative, associative, and distributive laws.
What is the golden rule of algebra?
- The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.
What are the 5 basic laws of algebra?
- The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.
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Microsoft Math Solver - Math Problem Solver & Calculator Get step-by-step solutions to your math problems Try Math Solver Type a math problem Solve Quadratic equation x2 − 4x − 5 = 0 Trigonometry 4sinθ cosθ = 2sinθ Linear equation y = 3x + 4 Arithmetic 699 ∗533 Matrix [ 2 5 3 4][ 2 −1 0 1 3 5] Simultaneous equation { 8x + 2y = 46 7x + 3y = 47
Type a math problem Solve algebra trigonometry Get step-by-step explanations See how to solve problems and show your work—plus get definitions for mathematical concepts Graph your math problems Instantly graph any equation to visualize your function and understand the relationship between variables Practice, practice, practice
To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
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You will need to get assistance from your school if you are having problems entering the answers into your online assignment. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. You may speak with a member of our customer support team by calling 1-800-876-1799.
QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose.
To solve your equation using the Equation Solver, type in your equation like x+4=5. The solver will then show you the steps to help you learn how to solve it on your own. Solving Equations Video Lessons Solving Simple Equations Need more problem types? Try MathPapa Algebra Calculator Show Keypad
Some equations are true for all allowed values and are then called Identities Example: sin (−θ) = −sin (θ) is one of the Trigonometric Identities Let's try θ = 30°: sin (−30°) = −0.5 and −sin (30°) = −0.5 So it is true for θ = 30° Let's try θ = 90°: sin (−90°) = −1 and −sin (90°) = −1 So it is also true for θ = 90° Is it true for all values of θ?
Two-Step Equation Word Problems: Integers Interpret this set of word problems that require two-step operations to solve the equations. Each printable worksheet has five word problems ideal for 6th grade, 7th grade, and 8th grade students. Two-Step Equation Word Problems: Fractions and Decimals
Section 2.2 : Linear Equations Solve each of the following equations and check your answer. 4x−7(2−x) = 3x+2 4 x − 7 ( 2 − x) = 3 x + 2 Solution 2(w+3)−10 = 6(32−3w) 2 ( w + 3) − 10 = 6 ( 32 − 3 w) Solution 4−2z 3 = 3 4 − 5z 6 4 − 2 z 3 = 3 4 − 5 z 6 Solution 4t t2−25 = 1 5 −t 4 t t 2 − 25 = 1 5 − t Solution
Problem 6 sent by Κυριάκος There is a two-digit number whose digits are the same, and has got the following property: When squared, it produces a four-digit number, whose first two digits are the same and equal to the original's minus one, and whose last two digits are the same and equal to the half of the original's.
For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems.
Equation Solver Step 1: Enter the Equation you want to solve into the editor. The equation calculator allows you to take a simple or complex equation and solve by best method possible. Step 2: Click the blue arrow to submit and see the result!
Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions. ... Algebra 1. Unit: Solving equations & inequalities. Not feeling ready for this? ... Using inequalities to solve problems Get 3 of 4 questions to level up! Compound inequalities. Learn ...
Solving equations & inequalities > Linear equations with variables on both sides Equations with variables on both sides CCSS.Math: 8.EE.C.7, 8.EE.C.7b Google Classroom Solve for f f. -f+2+4f=8-3f −f + 2+ 4f = 8 − 3f f = f = Stuck? Review related articles/videos or use a hint. Report a problem 7 4 1 x x y y \theta θ \pi π 8 5 2 0 9 6 3 Do 4 problems
An identity is an equation that is satisfied by all numbers from its replacement set. Example 1 Consider the equation 2x-1 = x+2. The replacement set here is the set of all real numbers. The equation is conditional since, for example, 1 is a member of the replacement set but not of the solution set. Example 2 Consider the equation (x-1) (x+1 ...
Given a general quadratic equation of the form ax²+bx+c=0 with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is: x= (-b±√ (b²-4ac))/2a where the plus-minus symbol "±" indicates that the quadratic equation has two solutions. Factorization
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; Exponential models; and Quadratic equations, functions, and graphs. Khan Academy's Algebra 1 course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!
Solving algebraic word problems requires us to combine our ability to create equations and solve them. To solve an algebraic word problem: Define a variable. Write an equation using the variable. Solve the equation. If the variable is not the answer to the word problem, use the variable to calculate the answer.
To obtain the solution for a word problem involving fractions, follow these steps: 1. State the problem clearly. 2. Add the phrase "show me the step-by-step solution" to your question. Simplify fractions without. To simplify fractions without showing the steps, utilize the phrase "solve this fraction" before posing your inquiries.
Learn about solve equations using our free math solver with step-by-step solutions.
How do you solve algebraic expressions? To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.
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