Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &x+3 y=12\\\ &-3 y+x=30 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 21\) and \(y = -3\).

Step-by-step solution

Rearrange the Equations

The first step is to rearrange the equations in ascending order means to write the \(x\) terms first then \(y\) terms. In this case, the equations can be written as \[x + 3y = 12\] and \[x - 3y = 30\]

Use the Linear Combination Method to Solve for One Variable

The linear combination method involves adding or subtracting the two equations to eliminate one of the variables. In this case, if we add the two equations, \[x + 3y + x - 3y = 12 + 30\], \(y\) terms cancel out, and we get \[2x = 42\]

Solve the Equation for \(x\)

By further simplifying the equation from Step 2, we can solve for \(x\). We can do this by dividing both sides of the equation by 2. That gives us \[x = 21\]

Substitute \(x\) into One of the Original Equations and Solve for \(y\)

By substituting \(x = 21\) into the first original equation \[x + 3y = 12\], we can solve for \(y\). Our equations becomes \[21 + 3y = 12\], which simplifies to \[3y = -9\], and further dividing by 3 gives us \[y = -3\]

Check the Solution

It's always important to check the solution by substituting the variables \(x = 21\) and \(y = -3\) back into the original equations to ensure they satisfy both equations, that is \[x + 3y = 21 - 3*-3 = 21 + 9 = 30\] and \[x - 3y = 21 - 3*-3 = 21 + 9 = 30\], which is true, so the solution is correct.