Solving a system of equations requires you to find the values of several variables in several equations. You can solve a system of equations through addition, subtraction, multiplication, or substitution. If you want to know how to solve a system of equations, just follow these steps.
Step
Method 1 of 4: Solving with Subtraction
Step 1. Write one equation on top of the other
Solving a system of equations by subtraction is a great way when you see that both equations have variables with the same coefficients with the same sign. For example, if both equations have a positive variable 2x, you should use the subtraction method to find the value of both variables.
- Write one equation on top of another by aligning the variables x and y and their whole numbers. Write the subtraction sign outside the quantity of the two systems of equations.
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Example: If your two equations are 2x + 4y = 8 and 2x + 27 = 2, then you should write the first equation above the second, with the sign of subtraction outside the quantity of the second system, indicating that you will be subtracting each part of the equation.
- 2x + 4y = 8
- -(2x + 2y = 2)
Step 2. Subtract equal parts
Now that you've aligned the two equations, all you have to do is subtract the equal parts. You can subtract the parts one by one:
- 2x - 2x = 0
- 4y - 2y = 2y
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8 - 2 = 6
2x + 4y = 8 -(2x + 2y = 2) = 0 + 2y = 6
Step 3. Do the rest
If you have eliminated one of the variables by getting an answer of 0 when you subtract variables with the same coefficient, you only need to solve the remaining variables by solving ordinary equations. You can omit 0 from the equation as it won't change its value.
- 2y = 6
- Divide 2y and 6 by 2 to get y = 3
Step 4. Plug the found value into one of the equations to find another value
Now that you know that y = 3, you just need to plug it into one of the original equations to find the value of x. It doesn't matter which equation you choose because the answer will be the same. If one equation looks more complicated than the other, just plug it into the simpler equation.
- Plug y = 3 into the equation 2x + 2y = 2 and find the value of x.
- 2x + 2(3) = 2
- 2x + 6 = 2
- 2x = -4
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x = - 2
You have solved the system of equations using subtraction. (x, y) = (-2, 3)
Step 5. Check your answers
To make sure that you solve the system of equations correctly, you can plug both your answers into both equations to make sure the answer is correct for both equations. Here's how to do it:
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Plug (-2, 3) for the value of (x, y) into the equation 2x + 4y = 8.
- 2(-2) + 4(3) = 8
- -4 + 12 = 8
- 8 = 8
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Plug (-2, 3) for the value of (x, y) into the equation 2x + 2y = 2.
- 2(-2) + 2(3) = 2
- -4 + 6 = 2
- 2 = 2
Method 2 of 4: Solving by Addition
Step 1. Write one equation on top of the other
Solving a system of equations by addition is the way to go if you see that both equations have variables with the same coefficients that have opposite signs. For example, if one of the equations has a variable of 3x and the other equation has a variable of -3x, then the addition method is the right way.
- Write one equation on top of another by aligning the variables x and y and their whole numbers. Write the addition sign outside the quantity of the second system of equations.
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Example: If your two equations are 3x + 6y = 8 and x – 6y = 4, then you should write the first equation above the second, with the addition sign outside the quantity of the second system, indicating that you will add up every part of the equation.
- 3x + 6y = 8
- +(x - 6y = 4)
Step 2. Add up the equal parts
Now that you've aligned the two equations, all you have to do is add up the equal parts. You can add them one by one:
- 3x + x = 4x
- 6y + -6y = 0
- 8 + 4 = 12
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When you combine them, you will get your new result:
- 3x + 6y = 8
- +(x - 6y = 4)
- = 4x + 0 = 12
Step 3. Do the rest
If you have eliminated one of the variables by getting 0 when you add up the variables with the same coefficient, you only need to solve the remaining variables by solving the ordinary equation. You can omit 0 from the equation as it won't change its value.
- 4x + 0 = 12
- 4x = 12
- Divide 4x and 12 by 3 to get x = 3
Step 4. Plug the result back into the equation to find another value
Now that you know that x = 3, you just need to plug it into one of the original equations to find the value of y. It doesn't matter which equation you choose because the result will be the same. If one equation looks more complicated than the other, just plug it into the simpler one.
- Plug x = 3 into the equation x – 6y = 4 to find the value of y.
- 3 - 6y = 4
- -6y = 1
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Divide -6y and 1 by -6 to get y = -1/6
You have solved the system of equations using addition. (x, y) = (3, -1/6)
Step 5. Check your answers
To make sure that you solve the system of equations correctly, you just need to plug the values into both equations to make sure that the answers to both equations are correct. Here's how to do it:
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Plug (3, -1/6) for the value (x, y) into the equation 3x + 6y = 8.
- 3(3) + 6(-1/6) = 8
- 9 - 1 = 8
- 8 = 8
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Plug (3, -1/6) for the value (x, y) into the equation x - 6y = 4.
- 3 - (6 * -1/6) =4
- 3 - - 1 = 4
- 3 + 1 = 4
- 4 = 4
Method 3 of 4: Solving by Multiplication
Step 1. Write one equation on top of the other
Write one equation on top of another by aligning the variables x and y and whole numbers. If you use the multiplication method, none of the variables have the same coefficient – not yet.
- 3x + 2y = 10
- 2x - y = 2
Step 2. Multiply one or both equations until one of the variables from both parts has the same coefficient
Now, multiply one or both equations by the same number which will make one of the variables have the same coefficient. In this problem, you can multiply the entire second equation by 2 so that the –y variable becomes -2y and equals the y coefficient of the first equation. Here's how to do it:
- 2 (2x - y = 2)
- 4x - 2y = 4
Step 3. Add or subtract the equations
Now, apply addition or subtraction to both equations using a method that will eliminate variables with the same coefficients. Since you want to solve 2y and -2y, you should use the addition method because 2y + -2y equals 0. If your problem is 2y and positive 2y, then you will use subtraction. Here's how to use the addition method to eliminate one of the variables:
- 3x + 2y = 10
- + 4x - 2y = 4
- 7x + 0 = 14
- 7x = 14
Step 4. Do the rest
Just solve it to find the value of the variable you didn't omit. If 7x = 14, then x = 2.
Step 5. Plug the value into the equation to find another value
Plug the value into one of the original equations to find the other. Choose a simpler equation to make it easier.
- 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)
Step 6. Check your answers
To check your answer, just plug the two values you found into the original equation to make sure you found the correct values.
- Plug (2, 2) for the value of (x, y) into the equation 3x + 2y = 10.
- 3(2) + 2(2) = 10
- 6 + 4 = 10
- 10 = 10
- Plug (2, 2) for the value of (x, y) into the equation 2x - y = 2.
- 2(2) - 2 = 2
- 4 - 2 = 2
- 2 = 2
Method 4 of 4: Solving with Substitution
Step 1. Align one of the variables
The substitution method is the correct way if one of the coefficients of one of the equations is equal to one. Then, all you have to do is isolate the coefficient of that one variable in one of the equations to find its value.
- If you're working on the equation 2x + 3y = 9 and x + 4y = 2, you'll want to isolate x in the second equation.
- x + 4y = 2
- x = 2 - 4y
Step 2. Plug the value of the variable that you have alone into another equation
Take the value you found when you isolated the variable and replace the variable in the equation that you didn't change with that value. You won't be able to solve anything if you plug it back into the equation you've changed. 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
Step 3. Solve the remaining variables
Now that you know that y = -1, just plug that value into a simpler equation to find the value of x. Here's how you do it:
- y = -1 x = 2 - 4y
- x = 2 - 4(-1)
- x = 2 - -4
- x = 2 + 4
- x = 6
- You have solved the system of equations by substitution. (x, y) = (6, -1)
Step 4. Check your work
To make sure that you're solving the system of equations correctly, you just need to plug your two answers into both equations to make sure that they're both correct. Here's how to do it:
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Plug (6, -1) for the value (x, y) into the equation 2x + 3y = 9.
- 2(6) + 3(-1) = 9
- 12 - 3 = 9
- 9 = 9
- Plug (6, -1) for the value (x, y) into the equation x + 4y = 2.
- 6 + 4(-1) = 2
- 6 - 4 = 2
- 2 = 2
