Question

Use substitution to I solve the linear system. Then use a graphing calculator or a computer to check your solution. $$\begin{aligned} &x-2 y=9\\\ &1.5 x+0.5 y=6.5 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 5\) and \(y = -2\).

Step-by-step solution

Rearrange one of the equations

Rearrange the first equation so that it is in terms of x, this will help us isolate the variable we want to solve for. From \(x - 2y = 9\), we can rearrange to \(x = 2y + 9\).

Substitute the rearranged equation into the other equation

Substitute \(x = 2y + 9\) from the first equation into the second equation. This gives us \(1.5(2y + 9) + 0.5y = 6.5\). Now this equation can be solved for y.

Solve for y

Solving for y, simplifies the equation to \(3y + 13.5 + 0.5y = 6.5\), combine like terms to get \(3.5y + 13.5 = 6.5\). Subtract 13.5 from both sides of the equation gives us \(3.5y = -7\), and dividing by 3.5 gives us \(y = -2\).

Substitute y = -2 back in to the first equation

Substitute \(y = -2\) into the first equation \(x=2(-2) + 9\) to find \(x = 5\).

Check the solution

Plug \(x = 5\) and \(y = -2\) into both original equations to confirm they hold true. This is an essential step to verify if the obtained values are the correct solutions.