Chapter 4, Problem 1RE

To determine

Whether the system of equations $\begin{array}{l}x-4y=-2\\ \text{2}x-8y=1\end{array}$ has exactly one solution, no solution, or an infinite number of solutions without graphing the equations and write equation each in the slope intercept form.

Answer to Problem 1RE

Solution:

The slope intercept form of the system of equation $\begin{array}{l}x-4y=-2\\ \text{2}x-8y=1\end{array}$ are $y=\frac{1}{4}x+\frac{2}{4},y=\frac{1}{4}x-\frac{1}{8}$.

The system of equations is inconsistent and has no solution.

Explanation of Solution

Given Information:

The provided system of linear equations is $\begin{array}{l}x-4y=-2\\ \text{2}x-8y=1\end{array}$.

Formula used:

The slope intercept form of the equation of line is,

$y=mx+c$

Where, m is the slope of the line and c is the y intercept of the line.

Consider the system of linear equations,

$\begin{array}{l}x-4y=-2\\ \text{2}x-8y=1\end{array}$

Consider the first equation $x-4y=-2$.

Rewrite the equation $x-4y=-2$ in the slope intercept form $y=mx+c$ as,

$\begin{array}{c}x-4y=-2\\ x+2=4y\\ y=\frac{x+2}{4}\\ y=\frac{1}{4}x+\frac{1}{2}\end{array}$

Compare the above equation with $y=mx+c$.

$m=\frac{1}{4}$ and $c=\frac{1}{2}$

Therefore, the slope of the line is $\frac{1}{4}$ and intercept is $\frac{1}{2}$.

Consider the second equation $\text{2}x-8y=1$.

Rewrite the equation $\text{2}x-8y=1$ in the slope intercept form $y=mx+c$.

$\begin{array}{c}\text{2}x-8y=1\\ 2x-1=8y\\ y=\frac{2x-1}{8}\\ y=\frac{1}{4}x-\frac{1}{8}\end{array}$

Compare the above equation with $y=mx+c$.

$m=\frac{1}{4}$ and $c=-\frac{1}{8}$

Therefore, the slope of the line is $\frac{1}{4}$ and intercept is $-\frac{1}{8}$.

Hence the slopes of both the lines is $\frac{1}{4}$ and the intercepts are $\frac{1}{2},-\frac{1}{8}$.

Since the lines are parallel as they have same slope and different intercepts. Therefore they do not intersect in a unique point, hence the system of equations is inconsistent and has no solution.

Thus, the system of equations is inconsistent and has no solution.