To solve a system of equations you have to find the value of more than one variable in more than one equation. It is possible to solve a system of equations using addition, subtraction, multiplication or substitution. If you want to learn how to solve a system of equations, follow the steps outlined in this article.
Steps
Method 1 of 4: Solve using Subtraction
Step 1. Write one equation above the other
Solving a system of equations by subtraction is ideal both equations have a variable with the same coefficient and the same sign. For example, if both equations have the positive variable 2x, it would be good to use the subtraction method to find the value of both variables.
- Write the equations on top of each other, aligning the x and y variables and the integers. Write the sign of the subtraction outside the parenthesis of the second equation.
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Ex: If the two equations are 2x + 4y = 8 and 2x + 2y = 2, you should write the first equation above the second, with the subtraction sign in front of the second equation, showing that you want to subtract each term of that equation.
- 2x + 4y = 8
- - (2x + 2y = 2)
Announce Your Retirement Step 8 Step 2. Subtract similar terms
Now that you've aligned the two equations, you just have to subtract the similar terms. You can do this by taking one term at a time:
- 2x - 2x = 0
- 4y - 2y = 2y
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8 - 2 = 6
2x + 4y = 8 - (2x + 2y = 2) = 0 + 2y = 6
Apply for an Entrepreneurial Grant Step 14 Step 3. Solve for the remaining term
Once you have eliminated one of the variables by subtracting the variables with the same coefficient, you can solve for the remaining variable by solving a normal equation. You can remove the 0 from the equation, since it will not change its value.
- 2y = 6
- Divide 2y and 6 by 2 to give y = 3
Stop Using Racist Comments Step 1 Step 4. Enter the term in one of the equations to find the value of the first term
Now that you know y = 3, you will need to substitute it in one of the initial equations to solve for x. No matter which equation you choose, the result will be the same. If one of the equations seems more difficult, choose the simpler equation.
- Substitute y = 3 in the equation 2x + 2y = 2 and solve for x.
- 2x + 2 (3) = 2
- 2x + 6 = 2
- 2x = -4
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x = - 2
You have solved the system of equations by subtraction. (x, y) = (-2, 3)
Defend Against Appropriation of Name or Likeness Claims Step 15 Step 5. Check the result
To make sure you have solved the system correctly, substitute the two results in both equations and verify that they are valid for both equations. Here's how to do it:
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Substitute (-2, 3) for (x, y) in the equation 2x + 4y = 8.
- 2(-2) + 4(3) = 8
- -4 + 12 = 8
- 8 = 8
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Substitute (-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: Solve with Addition
Study Late at Night Step 5 Step 1. Write one equation above the other
Solving a system of equations by addition is ideal when the two equations have a variable with the same coefficient and opposite sign. For example, if one equation has the variable 3x and the other has the variable -3x, then the addition method is ideal.
- Write the equations on top of each other, aligning the x and y variables and the integers. Write the plus sign outside the parenthesis of the second equation.
-
Ex: If the two equations are 3x + 6y = 8 and x - 6y = 4, you should write the first equation above the second, with the addition sign in front of the second equation, showing that you want to add each term of that equation.
- 3x + 6y = 8
- + (x - 6y = 4)
Calculate Profit Step 1 Step 2. Add the like terms
Now that you have aligned the two equations, you just have to add the similar terms together. You can do this by taking one term at a time:
- 3x + x = 4x
- 6y + -6y = 0
- 8 + 4 = 12
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When you combine it all, you will get:
- 3x + 6y = 8
- + (x - 6y = 4)
- = 4x + 0 = 12
Improve Your Life Step 5 Step 3. Solve for the remaining term
Once you have eliminated one of the variables by subtracting the variables with the same coefficient, you can solve for the remaining variable. You can remove the 0 from the equation, since it will not change its value.
- 4x + 0 = 12
- 4x = 12
- Divide 4x and 12 by 3 to give x = 3
Write a Grant Proposal Step 5 Step 4. Enter the term in the equation to find the value of the first term
Now that you know that x = 3, you will need to substitute it in one of the initial equations to solve for y. No matter which equation you choose, the result will be the same. If one of the equations seems more difficult, choose the simpler equation.
- Substitute x = 3 in the equation x - 6y = 4 and solve for y.
- 3 - 6y = 4
- -6y = 1
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Divide -6y and 1 by -6 to give y = -1/6
You have solved the system of equations by addition. (x, y) = (3, -1/6)
Write a Grant Proposal Step 17 Step 5. Check the result
To make sure you have solved the system correctly, substitute the two results in both equations and verify that they are valid for both equations. Here's how to do it:
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Substitute (3, -1/6) for (x, y) in the equation 3x + 6y = 8.
- 3(3) + 6(-1/6) = 8
- 9 - 1 = 8
- 8 = 8
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Substitute (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 with Multiplication
Write a Journal Step 3 Step 1. Write the equations on top of each other
Write the equations on top of each other, aligning the x and y variables and the integers. When using the multiplication method, the variables will still not have the same coefficients.
- 3x + 2y = 10
- 2x - y = 2
Overcome Boredom Step 1 Step 2. Multiply one or both equations until one of the variables of both terms has the same coefficient
Now, 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, so that the variable -y becomes -2y and has the same coefficient as the first y. Here's how to do it:
- 2 (2x - y = 2)
- 4x - 2y = 4
Write a Grant Proposal Step 12 Step 3. Add or subtract the equations
Now, use the addition or subtraction method in order to eliminate the variables that have the same coefficient. Since you are working with 2y and -2y, it would be better to use the addition method, since 2y + -2y equals 0. If you were working with 2y and 2y, then you should use the subtraction method. Here's how to use the addition method to delete one of the variables:
- 3x + 2y = 10
- + 4x - 2y = 4
- 7x + 0 = 14
- 7x = 14
Accept Mistakes and Learn from Them Step 6 Step 4. Solve for the remaining term
Solve to find the value of the term you didn't clear. If 7x = 14, then x = 2.
Deal With Different Problems in Life Step 17 Step 5. Enter the term in the equation to find the value of the first term
Insert the term into an original equation to solve for the other term. Choose the simplest equation to solve it more quickly.
- x = 2 - 2x - y = 2
- 4 - y = 2
- -y = -2
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y = 2
You have solved the system of equations with multiplication. (x, y) = (2, 2)
Define a Problem Step 10 Step 6. Check the result
To check the result, enter the two values into the original equations to make sure you have the right values.
- Substitute (2, 2) for (x, y) in the equation 3x + 2y = 10.
- 3(2) + 2(2) = 10
- 6 + 4 = 10
- 10 = 10
- Substitute (2, 2) for (x, y) in the equation 2x - y = 2.
- 2(2) - 2 = 2
- 4 - 2 = 2
- 2 = 2
Method 4 of 4: Solve using Substitution
Write a Book Report Step 3 Step 1. Isolate a variable
The substitution method is ideal when one of the coefficients of one of the equations is equal to one. What you need to do is isolate the variable with the single coefficient on one side of the equation and find its value.
- If you are working with the equations 2x + 3y = 9 and x + 4y = 2, it would be good to isolate x in the second equation.
- x + 4y = 2
- x = 2 - 4y
Accept Mistakes and Learn from Them Step 4 Step 2. Substitute the value of the variable you isolated into the other equation
Take the value found after isolating the variable and replace it in place of the variable in the equation that you have not changed. You won't be able to solve anything if you do the substitution in the same equation you just edited. Here's what to do:
- 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
Go to College with No Money Step 19 Step 3. Solve for the remaining variable
Now that you know that y = - 1, substitute its value in the easier equation to find x. Here's how to do it:
- y = -1 x = 2 - 4y
- x = 2 - 4 (-1)
- x = 2 - -4
- x = 2 + 4
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x = 6
You have solved the system of equations with substitution. (x, y) = (6, -1)
End a Letter Step 1 Step 4. Check your work
To make sure you have solved the system correctly, substitute the two results in both equations and verify that they are valid for both equations. Here's how to do it:
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Substitute (6, -1) for (x, y) in the equation 2x + 3y = 9.
- 2(6) + 3(-1) = 9
- 12 - 3 = 9
- 9 = 9
- Substitute (6, -1) for (x, y) in the equation x + 4y = 2.
- 6 + 4(-1) = 2
- 6 - 4 = 2
- 2 = 2
