Solving linear combinations
Do you have a particular example of a problem you need help with?
Sure! Let's say we have the following system of linear equations:
3x + 2y = 7
4x - y = 3
We can solve this system using linear combinations. Here's how:
1. Multiply one or both equations by a constant to make the coefficients of one of the variables the same (or opposites) in both equations. In this case, let's focus on the "y" variable. We can multiply the second equation by 2 to make the coefficients the same.
Equation 1: 3x + 2y = 7
Equation 2: 8x - 2y = 6
2. Add or subtract the two equations to eliminate one variable. In this case, if we add Equation 1 and Equation 2, the "y" terms will cancel out:
(3x + 2y) + (8x - 2y) = 7 + 6
11x = 13
3. Solve for the remaining variable. Divide both sides of the equation by 11:
x = 13/11
4. Substitute the value of x back into one of the original equations to solve for the other variable. Let's use Equation 1:
3(13/11) + 2y = 7
39/11 + 2y = 7
2y = 77/11 - 39/11
2y = 38/11
5. Finally, solve for y by dividing both sides by 2:
y = (38/11) / 2
y = 19/11
Therefore, the solution to the system of equations is x = 13/11 and y = 19/11.