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4.04 Equations with variables both sides


Separating letters and numbers

We have already looked at how to rearrange number values to solve equations. Now we're going to look at what to do when we have algebraic terms on both sides of the equation. Basically it's the same process. We want to group the like terms, so we have all the variables on one side and all the numbers on the other. Then we can solve the equation.


Worked example

Question 1

Solve the following equation: $6x-20=x$6x20=x

Think:  In order to solve for the variable, we need to isolate the variable on one side of the equation.  Since the variable appears on both sides of the equal sign, we will need to move one to the other side of the equation.


$6x-20$6x20 $=$= $x$x


$-6x+6x-20$6x+6x20 $=$= $-6x+x$6x+x

Subtract $6x$6x from both sides to remove $6x$6x from the left side of the equation.

$-20$20 $=$= $-5x$5x

Combine like terms

$\frac{-20}{-5}$205 $=$= $\frac{x}{-5}$x5

Divide both sides by $-5$5

$4$4 $=$= $x$x



Reflect:  It does not matter which side you choose to remove the variable from.  In the above example we chose to move all of the variables to the right hand side of the equal sign in order to avoid the right side of the equal sign becoming zero.  If, however, we chose to subtract $x$x from both sides as a first step, and we solved the resulting equation correctly, we would arrive at the exact same answer.  See the alternative solution below. Which do you prefer?

$6x-20$6x20 $=$= $x$x


$6x-x-20$6xx20 $=$= $x-x$xx

Subtract $x$x from both sides to remove $x$x from the right side of the equation.

$5x-20$5x20 $=$= $0$0

Combine like terms

$5x$5x $=$= $20$20

Add $20$20 to both sides

$\frac{5x}{5}$5x5 $=$= $\frac{20}{5}$205

Divide both sides by $5$5

$x$x $=$= $5$5



Practice questions

Question 2

Solve the following equation:


question 3

Solve the following equation for $x$x:


question 4

Solve the following equation:




Solve linear equations with rational number coefficients fluently, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Represent real-world problems using linear equations and inequalities in one variable and solve such problems.


Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by transforming a given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

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