Solving Linear Equations

by Justin Skycak (@justinskycak) on

Loosely speaking, a linear equation is an equality statement containing only addition, subtraction, multiplication, and division.

This post is part of the book Justin Math: Algebra. Suggested citation: Skycak, J. (2018). Solving Linear Equations. In Justin Math: Algebra. https://justinmath.com/solving-linear-equations/

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Loosely speaking, a linear equation is an equality statement containing only addition, subtraction, multiplication, and division. It does not need to include all of these operations, but it cannot include operations beyond them, such as exponentiation.

For example, these are linear equations:

5x+9=5
14x6=3x+2
5x+2+7x=3x+8x


On the other hand, these are not linear equations:

5x2+9=5
14x6=3x+2
5sin(x)+2+7x=3|x|+8


Solutions to Linear Equations

The solution of a linear equation is the value that we can substitute for the variable to make the equation true.

Most linear equations have a single solution. We can find the solution by performing operations on both sides of the equation, to isolate the variable.

Given equation|5x+8=2x+22Add 2x to both sides|7x+8=22Subtract 8 from both sides|7x=14Divide both sides by 7|x=2


To check our solution, we can substitute it in both sides of the equation and check that they evaluate to the same result:

5(2)+8=2(2)+2210+8=4+2218=18


Case of No Solutions

However, some linear equations have no solutions. When we try to solve these equations, the variable vanishes and we are left with an untrue statement.

Given equation|3x+1=2+3xSubtract 2x from both sides|1=2


This means that there is no number we can substitute for x to make the given equation true.

In fact, the right-hand side will always be 1 more than the left-hand side: the left-hand side says to multiply the input by 3 and add 1, while the left-hand side says to multiply the input by 3 and add 2.

Both sides multiply the input by 3, but then add different amounts! We can never hope to get the results to be the same.

Case of Infinitely Many Solutions

Even more interesting, some linear equations have infinitely many solutions. When we try to solve these equations, the variable still vanishes, but this time we are left with a true statement.

Given equation|2x+1=12xAdd 2x to both sides|1=1


In other words, any number we substitute for x will make the given equation true.

The left-hand side and the right-hand side will always come out to the same result: the left-hand side tells us to multiply the input by 2 and add 1, and the right-hand side tells us to multiply the input by 2 and then subtract it from 1. These are really just two ways of saying the same thing.

Exercises

Solve the following equations. (You can view the solution by clicking on the problem.)

1)2x+3=11
Solution:
x=4

2)3x+7=62x
Solution:
x=1

3)5+12=3x2x
Solution:
x=2

4)5x+2=2x17x
Solution:
no solution

5)3x1712x=5x+13
Solution:
x=3

6)18x+1=10x+1911x
Solution:
infinitely many solutions

7)9(x+4)+12(x4)=84
Solution:
x=5

8)4(2x+3)x=3(2x+4)+x
Solution:
infinitely many solutions

9)5[4(3x+1)3(3x+2)]+25=75
Solution:
x=6

10)3[2(x+1)(x+2)]=3x+1
Solution:
no solution


This post is part of the book Justin Math: Algebra. Suggested citation: Skycak, J. (2018). Solving Linear Equations. In Justin Math: Algebra. https://justinmath.com/solving-linear-equations/


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