Question

Use substitution to solve the linear system. $$\begin{aligned} &3 x+y=3\\\ &7 x+2 y=1 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = -5\) and \(y = 18\).

Step-by-step solution

Express one variable in terms of the other from one of the equations

Let's take the first equation \(3x + y = 3\), and express \(y\) in terms of \(x\): \(y = 3 - 3x\). This simplification is gotten by subtracting \(3x\) from both sides.

Substitute the solved equation into the other equation

Now substitute \(y = 3 - 3x\) into the second equation \(7x + 2y = 1\): \(7x + 2(3 - 3x) = 1\).

Solve for x

On expanding and simplifying the equation from Step 2, we get: \(7x + 6 - 6x = 1\), which simplifies to \(x = -5\).

Substitute x into the solved equation for y

Substitute \(x = -5\) into the equation \(y = 3 - 3x\): \(y = 3 - 3(-5)\).

Solve for y

On simplifying the equation from Step 4, we get: \(y = 18\).

Solution verification

Substitute \(x = -5\) and \(y = 18\) into the original equations to verify the solution. Both original equations yield true statements, meaning our solution is correct.