How to Solve Equations with Variables on Both Sides

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The basic properties of solving equations with variables on both sides

Co-authored by JohnK Wright V

Last Updated: December 13, 2024 Fact Checked

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Equations with One Variable on Both Sides

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System Equations with Two Variables

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Example Problems

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Q&A

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To study algebra, you will see equations that have a variable on one side, but later on you will often see equations that have variables on both sides. The most important thing to remember when solving such equations is that whatever you do to one side of the equation, you must do to the other side. Using this rule, it is easy to move variables around so that you can isolate them and use basic operations to find their value.

How to Solve for X on Both Sides

Cancel out parenthesis by multiplying each term in the parenthesis by outside term.
Combine all the terms with the same variable.
Add your variables together on one side and subtract them from the other.
Isolate variables by dividing them by their constants.
Remove coefficients by applying the inverse root to the variable.
Balance your equation by doing each action to both sides.
Move your constants to the other side, if necessary.

Part 1

Part 1 of 3:

Equations with One Variable on Both Sides

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1
Apply the distributive property, if necessary. The distributive property states that $a(b+c)=ab+ac$ . ^{[1]
X
Research source} This rule allows you to cancel out parentheses by multiplying each term in the parentheses by the number outside the parentheses. ^{[2]
X
Research source}
- For example, if your equation is $2(10-2x)=4(2x+2)$ , use the distributive property to multiply the terms in parentheses by the number outside the parentheses:
  $2(10-2x)=4(2x+2)$
  $20-4x=8x+8$
2
Cancel the variable on one side of the equation. To cancel the variable, complete the opposite operation as stated in the equation. For example, if the term is subtracted in the equation, cancel it by adding. If the term is added in the equation, cancel it by subtracting. It is usually easiest to cancel the variable with the smaller coefficient.
- For example, in the equation $20-4x=8x+8$ , cancel the term $-4x$ by adding $4x$ :
  $20-4x+4x=8x+8$ .
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3
Keep the equation balanced. Whatever you do to one side of the equation, you must do to the other side as well. So if you add or subtract to cancel the variable on one side of the equation, you must add or subtract to the other side as well.
- For example, if you added $4x$ on one side of the equation to cancel the variable, you must also add $4x$ to the other side of the equation:
  $20-4x+4x=8x+8+4x$
4
Simplify the equation by combining like terms. You should now have the variable on one side of the equation.
- For example:
  $20-4x+4x=8x+8+4x$
  $20=12x+8$
5
Move the constants to one side of the equation, if necessary. You want the variable term on one side, and the constant on the other side. To move the constant to one side, add or subtract from each side of the equation to cancel the term on one side. ^{[3]
X
Research source}
- For example, to cancel the $+8$ constant on the variable side, subtract 8 from both sides of the equation:
  $20=12x+8$
  $20-8=12x+8-8$
  $12=12x$
6
Cancel the variable’s coefficient. To do this, perform the operation opposite from the one denoted in the equation. Usually this will mean dividing to cancel a coefficient being multiplied by a variable. ^{[4]
X
Research source} Remember that whatever you do to one side of the equation, you must do to the other side of the equation as well.
- For example, to cancel out the coefficient 12 from the equation, you would divide each side of the equation by 12:
  $12=12x$
  ${\frac {12}{12}}={\frac {12x}{12}}$
  $1=x$
7
Check your work. To make sure your answer is correct, substitute your solution back into the original equation. If the equation is true, your answer is correct.
- For example, if $1=x$ , substitute 1 for the variable in the equation and calculate:
  $2(10-2x)=4(2x+2)$
  $2(10-2(1))=4(2(1)+2)$
  $2(10-2)=4(2+2)$
  $20-4=8+8$
  $16=16$

Part 2

Part 2 of 3:

System Equations with Two Variables

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1
Isolate a variable in one equation. This might already be done. If not, use the rules of algebra to isolate the variable on one side of the equation. Remember that whatever you do to one side of the equation, you must do to the other side.
- For example, for the equation $y+1=x-1$ , to isolate the $y$ variable, you would subtract 1 from both sides:
  $y+1=x-1$
  $y+1-1=x-1-1$
  $y=x-2$
EXPERT TIP

Joseph Meyer
Math Teacher
Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.

Joseph Meyer
Math Teacher

To solve an equation for a variable like "x," you need to manipulate the equation to isolate x. Use techniques like the distributive property, combining like terms, factoring, adding or subtracting the same number, and multiplying or dividing by the same non-zero number to isolate "x" and find the answer.
2
Substitute the value of the isolated variable into the other equation. Make sure you substitute the entire expression for the variable. This will give you an equation with only one variable, allowing you to solve for the variable. ^{[5]
X
Research source}
- For example, if your first equation is $2x=20-2y$ , and you determined $y=x-2$ in the second equation, you would substitute $x-2$ for $y$ in the first equation:
  $2x=20-2y$
  $2x=20-2(x-2)$
3
Solve for the variable. To do this, move the variable to one side of the equation. Then, move the constants to one side of the equation. Then, isolate the variable using multiplication or division.
- For example:
  $2x=20-2(x-2)$
  $2x=20-2x+4$
  $2x=24-2x$
  $2x+2x=24-2x+2x$
  $4x=24$
  ${\frac {4x}{4}}={\frac {24}{4}}$
  $x=6$
4
Solve for the remaining variable. To do this, plug the value of the variable you already solved into one of the equations. This will give you an equation with only one variable. Solve for the variable using the rules of algebra. You can use either equation to solve for the remaining variable.
- For example, if you found that $x=6$ , you can substitute 6 for $x$ in the second equation:
  $y=x-2$
  $y=(6)-2$
  $y=4$
5
Check your work. Plug the values for both variables into one of the equations. If the equation is true, your solutions are correct.
- For example, if you found that $x=6$ and $y=4$ , plug these back into the original equation and solve:
  $2x=20-2y$
  $2(6)=20-2(4)$
  $12=20-8$
  $12=12$

Part 3

Part 3 of 3:

Example Problems

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1
Try this problem using the distributive property with one variable: $5(x+4)=6x-5$ .
- Use the distributive property to cancel the parentheses:
  $5(x+4)=6x-5$
  $5x+20=6x-5$
- Cancel the $5x$ on the left side of the equation by subtracting $5x$ from both sides:
  $5x+20=6x-5$
  $5x+20-5x=6x-5-5x$
  $20=x-5$
- Isolate the variable by adding 5 to each side of the equation:
  $20=x-5$
  $20+5=x-5+5$
  $25=x$
EXPERT TIP

Joseph Meyer
Math Teacher
Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.

Joseph Meyer
Math Teacher

The distributive property helps you avoid repetitive calculations. You can use the distributive property to solve equations where you must multiply a number by a sum or difference. It simplifies calculations, enables expression manipulation (like factoring), and forms the basis for solving many equations.
2
Try this problem involving a fraction: $-7+3x={\frac {7-x}{2}}$ .
- Remove the fraction. To do this, multiply each side of the equation by the fraction’s denominator:
  $-7+3x={\frac {7-x}{2}}$
  $2(-7+3x)=2({\frac {7-x}{2}})$
  $-14+6x=7-x$
- Cancel the $-x$ on the right side of the equation by adding $x$ to each side of the equation:
  $-14+6x=7-x$
  $-14+6x+x=7-x+x$
  $-14+7x=7$
- Move the constants to one side of the equation by adding 14 to each side:
  $-14+7x=7$
  $-14+7x+14=7+14$
  $7x=21$
- Cancel the coefficient by dividing each side of the equation by 7:
  $7x=21$
  ${\frac {7x}{7}}={\frac {21}{7}}$
  $x=3$
3
Try solving this system of equations: $9x+15=12y;9y=9x+27$
- Isolate the $y$ variable in the second equation:
  $9y=9x+27$
  $9y=9(x+3)$
  ${\frac {9y}{9}}={\frac {9(x+3)}{9}}$
  $y=x+3$
- Plug in $x+3$ for $y$ in the first equation:
  $9x+15=12y$
  $9x+15=12(x+3)$
- Use the distributive property to cancel the parentheses:
  $9x+15=12x+36$
- Cancel the variable on the left side of the equation by subtracting $9x$ from each side:
  $9x+15=12x+36$
  $9x+15-9x=12x+36-9x$
  $15=3x+36$
- Move the constants to one side by subtracting 36 from each side:
  $15=3x+36$
  $15-36=3x+36-36$
  $-21=3x$
- Cancel the coefficient by dividing each side by 3:
  $-21=3x$
  ${\frac {-21}{3}}={\frac {3x}{3}}$
  $-7=x$
- Solve for $y$ by plugging the value of $x$ into either equation:
  $9y=9x+27$
  $9y=9(-7)+27$
  $9y=-63+27$
  $9y=-36$
  ${\frac {9y}{9}}={\frac {-36}{9}}$
  $y=-4$

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Question

How do I solve 6z = -54?

Community Answer

Solving for z, you would divide -54 by six to get singular z, with the end result being z=-9.

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How do I solve 3y + 9 = 9x -18? It's hard for me because they're two different variables.

Donagan

Top Answerer

You solve for either variable in terms of the other variable (meaning that y will be equal to some expression with x in it, or that x will be equal to some expression with y in it). In this problem you could start by subtracting 9 from both sides of the equation, giving you 3y = 9x - 27. Then to solve for y, divide both sides of the equation by 3. To solve for x instead, first add 27 to both sides of the equation and then divide by 9.

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How would I solve 3y + 9 = 9x - 18?

Donagan

Top Answerer

Decide whether you want to solve for x in terms of y or vice versa. If solving for x, divide both sides of the equation by 9. That gives you (y/3) + 1 = x - 2. Add 2 to both sides: (y/3) + 3 = x. If solving for y, divide both sides by 3, which gives you y + 3 = 3x - 6. Subtract 3 from both sides: y = 3x - 9.

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How to Solve Equations with Variables on Both Sides

How to Solve for X on Both Sides

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Equations with One Variable on Both Sides

System Equations with Two Variables

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