For example, if both equations have the variable positive 2x, you should use the subtraction method to find the value of both variables. Write one equation above the other by matching up the x and y variables and the whole numbers. Write the subtraction sign outside the quantity of the second system of equations. Ex: If your two equations are 2x + 4y = 8 and 2x + 2y = 2, then you should write the first equation over the second, with the subtraction sign outside the quantity of the second system, showing that you’ll be subtracting each of the terms in that equation. 2x + 4y = 8 -(2x + 2y = 2)
2x - 2x = 0 4y - 2y = 2y 8 - 2 = 6 2x + 4y = 8 -(2x + 2y = 2) = 0 + 2y = 6
2y = 6 Divide 2y and 6 by 2 to get y = 3
Plug 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)
Plug (-2, 3) in for (x, y) in the equation 2x + 4y = 8. 2(-2) + 4(3) = 8 -4 + 12 = 8 8 = 8 Plug (-2, 3) in for (x, y) in the equation 2x + 2y = 2. 2(-2) + 2(3) = 2 -4 + 6 = 2 2 = 2
Write one equation above the other by matching up the x and y variables and the whole numbers. Write the addition sign outside the quantity of the second system of equations. Ex: If your two equations are 3x + 6y = 8 and x - 6y = 4, then you should write the first equation over the second, with the addition sign outside the quantity of the second system, showing that you’ll be adding each of the terms in that equation. 3x + 6y = 8 +(x - 6y = 4)
3x + x = 4x 6y + -6y = 0 8 + 4 = 12 When you combine it all together, you get your new product: 3x + 6y = 8 +(x - 6y = 4) = 4x + 0 = 12
4x + 0 = 12 4x = 12 Divide 4x and 12 by 3 to get x = 3
Plug x = 3 into the equation x - 6y = 4 to solve for y. 3 - 6y = 4 -6y = 1 Divide -6y and 1 by -6 to get y = -1/6 You have solved the system of equations by addition. (x, y) = (3, -1/6)
Plug (3, -1/6) in for (x, y) in the equation 3x + 6y = 8. 3(3) + 6(-1/6) = 8 9 - 1 = 8 8 = 8 Plug (3, -1/6) in for (x, y) in the equation x - 6y = 4. 3 - (6 * -1/6) =4 3 - - 1 = 4 3 + 1 = 4 4 = 4
3x + 2y = 10 2x - y = 2
2 (2x - y = 2) 4x - 2y = 4
3x + 2y = 10 + 4x - 2y = 4 7x + 0 = 14 7x = 14
x = 2 —> 2x - y = 2 4 - y = 2 -y = -2 y = 2 You have solved the system of equations by multiplication. (x, y) = (2, 2)
Plug (2, 2) in for (x, y) in the equation 3x + 2y = 10. 3(2) + 2(2) = 10 6 + 4 = 10 10 = 10 Plug (2, 2) in for (x, y) in the equation 2x - y = 2. 2(2) - 2 = 2 4 - 2 = 2 2 = 2
If you’re working with the equations 2x + 3y = 9 and x + 4y = 2, you should isolate x in the second equation. x + 4y = 2 x = 2 - 4y
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
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)
Plug (6, -1) in for (x, y) in the equation 2x + 3y = 9. 2(6) + 3(-1) = 9 12 - 3 = 9 9 = 9 Plug (6, -1) in for (x, y) in the equation x + 4y = 2. 6 + 4(-1) = 2 6 - 4 = 2 2 = 2
