Content

• To step
• Method 1 of 4: Solve by Subtraction
• Method 2 of 4: Solving by Addition
• Method 3 of 4: Solve by multiplying
• Method 4 of 4: Dissolve by Substitution
• Tips

Solving a system of equations requires finding the value of multiple variables in multiple equations. You can solve a system of equations using addition, subtraction, multiplication or substitution. If you would like to know how to solve a system of equations, all you have to do is follow these steps.

To step

Method 1 of 4: Solve by Subtraction

1. Write one equation on top of the other. Solving these equations with subtraction is an ideal method when you see that both equations have the same variable with the same coefficient and the same sign. For example, if both equations have the variable -2x, you can use subtraction to find the value of both variables.
   • Write one equation on top of the other so that the x and y variables of both equations and the numbers are one below the other. Place the minus sign next to the bottom number.
   • Ex: If you have the following two equations: 2x + 4y = 8 and 2x + 2y = 2, it looks like this:
     • 2x + 4y = 8
     • - (2x + 2y = 2)
2. Subtract like terms. Now that the two equations are aligned, all you have to do is subtract the like terms. Do this with one term at a time:
   • 2x - 2x = 0
   • 4y - 2y = 2y
   • 8 - 2 = 6
     • 2x + 4y = 8 - (2x + 2y = 2) = 0 + 2y = 6
3. Solve for the remaining term. Remove any zero from the resulting equation, it doesn't change the value, and solve for the remaining equation.
   • 2y = 6
   • Divide 2y and 6 by 2 to get y = 3
4. Enter the found value of the variable in one of the equations. Now that you know that y = 3, you can enter this value into the original equation to solve for x. No matter which equation you choose, the answer is the same. So use the simplest equation!
   • Enter y = 3 into the equation 2x + 2y = 2 and solve for x.
   • 2x + 2 (3) = 2
   • 2x + 6 = 2
   • 2x = -4
   • x = - 2
     • You have solved the system of equations by subtraction. (x, y) = (-2, 3)
5. Check your answer. To make sure your answer is correct, enter both answers in both equations. Here you can see how:
   • Enter (-2, 3) for (x, y) in the equation 2x + 4y = 8.
     • 2(-2) + 4(3) = 8
     • -4 + 12 = 8
     • 8 = 8
   • Enter (-2, 3) for (x, y) in the equation 2x + 2y = 2.
     • 2(-2) + 2(3) = 2
     • -4 + 6 = 2
     • 2 = 2

Method 2 of 4: Solving by Addition

1. Write one equation on top of the other. Solving a system of equations by addition is the best method if you notice that both equations have a variable with the same coefficient, but with a different sign; for example, if one equation contains the variable 3x and the other contains the variable -3x.
   • Write one equation on top of the other so that the x and y variables of both equations and the numbers are one below the other. Place the plus sign next to the bottom number.
   • Ex: You have the following two equations 3x + 6y = 8 and x - 6y = 4, then write the first equation above the second as shown below:
     • 3x + 6y = 8
     • + (x - 6y = 4)
2. Add similar terms together. Now that the two equations are aligned, all you have to do is add the terms with the same variable:
   • 3x + x = 4x
   • 6y + -6y = 0
   • 8 + 4 = 12
   • If you combine these you will get a new product:
     • 3x + 6y = 8
     • + (x - 6y = 4)
     • = 4x ​​+ 0 = 12
3. Solve for the remaining term. Remove any zero from the resulting equation, it does not change the value. Solve the remaining equation.
   • 4x + 0 = 12
   • 4x = 12
   • Divide 4x and 12 by 3 to get x = 3
4. Enter the found value of this variable in one of the equations. Now that you know that x = 3, you can enter this value into the original equation to solve for y. No matter which equation you choose, the answer is the same. So use the simplest equation!
   • Enter x = 3 into the equation x - 6y = 4 to find y.
   • 3 - 6y = 4
   • -6y = 1
   • Divide -6y and 1 by -6 to get y = -1/6.
     • You have solved the system of equations with addition. (x, y) = (3, -1/6)
5. Check your answer. To make sure your answer is correct, enter both answers in both equations. Here's how:
   • Enter (3, -1/6) for (x, y) in the equation 3x + 6y = 8.
     • 3(3) + 6(-1/6) = 8
     • 9 - 1 = 8
     • 8 = 8
   • Enter (3, -1/6) for (x, y) in the equation x - 6y = 4.
     • 3 - (6 * -1/6) =4
     • 3 - - 1 = 4
     • 3 + 1 = 4
     • 4 = 4

Method 3 of 4: Solve by multiplying

1. Write one equation on top of the other. Write one equation on top of the other so that the x and y variables of both equations and the numbers are one below the other. If you're using multiplication, you're doing it because none of the variables have equal coefficients - right now.
   • 3x + 2y = 10
   • 2x - y = 2
2. Provide equal coefficients. Then multiply one or both equations by a number, so that one of the variables has the same coefficient. In this case, you can multiply the entire second equation by 2 to make -y equal to -2y and thus the first y coefficient. Here's how to do that:
   • 2 (2x - y = 2)
   • 4x - 2y = 4
3. Add or subtract the equations. Now all you have to do is eliminate similar terms by adding or subtracting. Since you are dealing with 2y and -2y here, it makes sense to use the addition method as it equals 0. If you are dealing with 2y + 2y, use the subtraction method. Here's an example of how to use the addition method to cancel variables:
   • 3x + 2y = 10
   • + 4x - 2y = 4
   • 7x + 0 = 14
   • 7x = 14
4. Solve this for the remaining term. This is easily solved by finding the value of the term that you have not yet eliminated. If 7x = 14, then x = 2.
5. Enter the value found in one of the equations. Enter the term in one of the original equations to solve for the other term. Choose the simplest equation for this, this is the fastest.
   • x = 2 ---> 2x - y = 2
   • 4 - y = 2
   • -y = -2
   • y = 2
   • You have solved the system of equations using multiplication. (x, y) = (2, 2)
6. Check your answer. To make sure your answer is correct, enter both answers in both equations. Here you can see how:
   • Enter (2, 2) for (x, y) in the equation 3x + 2y = 10.
   • 3(2) + 2(2) = 10
   • 6 + 4 = 10
   • 10 = 10
   • Enter (2, 2) for (x, y) in the equation 2x - y = 2.
   • 2(2) - 2 = 2
   • 4 - 2 = 2
   • 2 = 2

Method 4 of 4: Dissolve by Substitution

1. Isolate a variable. Substitution is ideal when one of the coefficients in one of the equations equals 1. Then all you have to do is isolate this variable on one side of the equation to find its value.
   • If you are working with the equations 2x + 3y = 9 and x + 4y = 2, you have to isolate x in the second equation.
   • x + 4y = 2
   • x = 2 - 4y
2. Enter the value of the variable you isolated in the other equation. Take the value of the isolated variable and fill it in the other equation. Of course not in the same comparison, otherwise you will not solve anything. Here's an example of how to do that:
   • x = 2 - 4y -> 2x + 3y = 9
   • 2 (2 - 4y) + 3y = 9
   • 4 - 8y + 3y = 9
   • 4 - 5y = 9
   • -5y = 9 - 4
   • -5y = 5
   • -y = 1
   • y = -1
3. Solve for the remaining variable. Now that you know that y = - 1, enter this value into the simpler equation to find the value of x. Here's an example of how to do that:
   • y = -1 -> x = 2 - 4y
   • x = 2 - 4 (-1)
   • x = 2 - -4
   • x = 2 + 4
   • x = 6
   • You have solved the system of equations using substitution. (x, y) = (6, -1)
4. Check your answer. To make sure your answer is correct, enter both answers in both equations. Here you can see how:
   • Enter (6, -1) for (x, y) in the equation 2x + 3y = 9.
     • 2(6) + 3(-1) = 9
     • 12 - 3 = 9
     • 9 = 9
   • Enter (6, -1) for (x, y) in the equation x + 4y = 2.
   • 6 + 4(-1) = 2
   • 6 - 4 = 2
   • 2 = 2

Tips

• You should now be able to solve any linear system of equations using addition, subtraction, multiplication, or substitution, but one method is usually best, depending on the equations.