Question

2. What is the solution for the system of equations?16x - 32y = 278x - 16 = 16ya) Use the linear combination (elimination) method to solve the system of equations.b) What does the solution tell you about the two lines of the system?

Answer 1

The given system of equations is:

$\begin{cases}16x-32y={27} \\ 8x-16={16y}\end{cases}$

It is required to use the elimination method to solve the system of equations and then state what the solution tells about the two lines of the system.

(a) To do this, express the variable x in the second equation in terms of y and substitute it into the first equation:

$\begin{gathered} 8x-16=16y \\ \text{ Add }16\text{ to both sides:} \\ \Rightarrow8x-16+16=16y+16 \\ \Rightarrow8x=16y+16 \\ \text{ Divide both sides by }8: \\ \Rightarrow\frac{8x}{8}=\frac{16y}{8}+\frac{16}{8} \\ \Rightarrow x=2y+2 \end{gathered}$

Substitute into the first equation:

$\begin{gathered} 16(2y+2)-32y=27 \\ \Rightarrow32y+32-32y=27 \\ \Rightarrow32=27 \end{gathered}$

Notice that the simplification results in a false equation.

Hence, the system of equations has no solution.

(b) Since the system has no solution, it follows that the lines of the system do not intersect.

This implies that the two lines are parallel lines.