Solving a system of equations amounts to finding the value of several unknowns using several equations. You can solve a system of equations by addition, subtraction, multiplication, or substitution. If you want to know how to solve a system of equations, just follow these steps.
Steps
Method 1 of 4: Solving by subtraction
Step 1. Write the equations one under the other
You can use the subtraction method when both equations have an unknown with the same coefficient and the same sign.
- For example, if both equations contain 2x, you need to use the subtraction method to find the value of x and y.
- Write the equations on top of each other, aligning the xs, ys, and constants. Put the subtraction sign to the left of the second equation.
- Example: if your two equations are 2x + 4y = 8 and 2x + 2y = 2, then you need to align the two equations vertically, with the subtraction sign to the left of the second equation, thus meaning that you are subtracting the two equations from term to term:
- 2x + 4y = 8
- - (2x + 2y = 2)
Step 2. Subtract term by term
Now that you've got the two equations aligned, all you need to do is subtract similar terms. You can operate term after term as follows:
- 2x - 2x = 0
- 4y - 2y = 2y
- 8 - 2 = 6
2x + 4y = 8 - (2x + 2y = 2) = 0 + 2y = 6
Step 3. Find the other stranger
Once you've eliminated one of the two unknowns, you just need to find the other unknown (here, y). Take the 0 out of the equation, because it's no use.
- 2y = 6
- y = 6/2, or y = 3
Step 4. Numerically apply one of the equations to find the value of the first unknown
Now that you know that y = 3, you just need to numerically apply one of the equations to find x. No matter which equation you choose, the result will be the same. If one of the equations seems more complicated than the other, choose the simpler.
- Numerically apply y = 3 of the equation 2x + 2y = 2 to find x.
- 2x + 2 (3) = 2
- 2x + 6 = 2
- 2x = -4
- x = - 2
You have solved the system of equations by subtraction. The answer is therefore the couple: (x, y) = (-2, 3)
Step 5. Check your answer
To make sure that you have correctly solved your system of equations, do the numerical application with both solutions in both equations to make sure that it works. Here's how to do it.
- Do the numerical application with (x, y) = (-2, 3) from the equation 2x + 4y = 8.
- 2(-2) + 4(3) = 8
- -4 + 12 = 8
- 8 = 8
- Do the numerical application with (x, y) = (-2, 3) from 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 the equations one under the other
You can use the addition method when the two equations have an unknown with the same coefficient, but opposite signs. For example, if one of the two equations contains 3x, and the other contains -3x.
- Write the equations on top of each other, aligning the xs, ys, and constants. Put the addition sign to the left of the second equation.
- Example: if your two equations are 3x + 6y = 8 and x - 6y = 4, then you need to align the two equations vertically, with the addition sign to the left of the second equation, thus meaning that you add the two equations term in the long term:
- 3x + 6y = 8
- + (x - 6y = 4)
Step 2. Add term by term
Now that you've got the two equations aligned, all you need to do is add up the similar terms. You can operate term after term as follows:
- 3x + x = 4x
- 6y + -6y = 0
- 8 + 4 = 12
- you then get:
- 3x + 6y = 8
- + (x - 6y = 4)
- = 4x + 0 = 12
Step 3. Find the other stranger
Once you've eliminated one of the two unknowns, you just need to find the other unknown (here, y). Take the 0 out of the equation, because it's no use.
- 4x + 0 = 12
- 4x = 12
- x = 12/4, or x = 3
Step 4. Numerically apply one of the equations to find the value of the first unknown
Now that you know that x = 3, you just need to numerically apply one of the equations to find x. No matter which equation you choose, the result will be the same. If one of the equations seems more complicated than the other, choose the simpler.
- Numerically apply x = 3 of the equation x - 6y = 4 to find y.
- 3 - 6y = 4
- -6y = 1
- y = 1 / -6, or y = -1/6
You have solved the system of equations by addition. The answer is therefore the couple: (x, y) = (3, -1/6)
Step 5. Check your answer
To make sure that you have correctly solved your system of equations, do the numerical application with both solutions in both equations to make sure that it works. Here's how to do it.
- Do the numerical application with (x, y) = (3, 1/6) of the equation 3x + 6y = 8.
- 3(3) + 6(-1/6) = 8
- 9 - 1 = 8
- 8 = 8
- Do the numerical application with (x, y) = (3, 1/6) of 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 the equations one under the other
Write the equations on top of each other, aligning the xs, ys, and constants. We use the multiplication method when the unknowns have different coefficients … for now!
- 3x + 2y = 10
- 2x - y = 2
Step 2. Multiply one of the two equations, or both, until one of the unknowns has the same coefficient in both equations
Now multiply either or both of the equations by a number so that one of the unknowns has the same coefficient in both equations. In our case, we can multiply the second equation by 2, so that -y becomes -2y, unknown that we have in the first equation with the same coefficient. Which give:
- 2 (2x - y = 2)
- 4x - 2y = 4
Step 3. Add or subtract the two equations
Now, it suffices to use either the addition method or the subtraction method in order to eliminate one of the two unknowns. As we have 2y and -2y in our case, we will use the addition method, because 2y + -2y is equal to 0. If you had 2y and 2y, we would have used the subtraction method. Apply here the addition method to eliminate y:
- 3x + 2y = 10
- + 4x - 2y = 4
- 7x + 0 = 14
- 7x = 14
Step 4. Find the other stranger
Solve this simple equation. If 7x = 14, then x = 2.
Step 5. Do the numeric application with x = 2 to find the value of the other unknown
Numerically apply one of the equations to find y. No matter which equation you choose, the result will be the same. If one of the equations seems more complicated than the other, choose the simpler.
- x = 2 - 2x - y = 2
- 4 - y = 2
- -y = -2
- y = 2
- You have solved the system of equations by multiplication. The answer is therefore the couple: (x, y) = (2, 2)
Step 6. Check your answer
To make sure that you have correctly solved your system of equations, do the numerical application with both solutions in both equations to make sure that it works. Here's how to do it.
- Do the numerical application with (x, y) = (2, 2) of the equation 3x + 2y = 10.
- 3(2) + 2(2) = 10
- 6 + 4 = 10
- 10 = 10
- Do the numerical application with (x, y) = (2, 2) of the equation 2x - y = 2.
- 2(2) - 2 = 2
- 4 - 2 = 2
- 2 = 2
Method 4 of 4: Solving by substitution
Step 1. Isolate one of the unknowns
The substitution method works well when one of the unknowns has a coefficient of 1 in one of the two equations. Then all you have to do is isolate that unknown.
- If your two equations are: 2x + 3y = 9 and x + 4y = 2, isolate x in the second equation.
- x + 4y = 2
- x = 2 - 4y
Step 2. Do the numerical application in the second equation with this unknown factor that you have just isolated
Replace the x value in the second equation with the x value you isolated. Be careful not to do the application with the first equation, which would be pointless! Which give:
- 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. Find the other stranger
Since y = - 1, do the numerical map in one of the starting equations to find x. Which give:
- 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. The answer is therefore the couple: (x, y) = (6, -1)
Step 4. Check your answer
To make sure that you have correctly solved your system of equations, do the numerical application with both solutions in both equations to make sure that it works. Here's how to do it.
- Do the numerical application with (x, y) = (6, -1) from the equation 2x + 3y = 9.
- 2(6) + 3(-1) = 9
- 12 - 3 = 9
- 9 = 9
- Do the numerical application with (x, y) = (6, -1) of the equation x + 4y = 2.
- 6 + 4(-1) = 2
- 6 - 4 = 2
- 2 = 2
