Write the following equations in slope-intercept form. Then, without solving the system, determine how many solutions the system has.

[tex]\[
\begin{aligned}
5x & = -2y + 5 \\
20x & = -8y + 20
\end{aligned}
\][/tex]

The first equation in slope-intercept form is [tex]$\square$[/tex]. (Simplify your answer. Use integers or fractions for any numbers in the expression.)



Answer :

To write the equations in slope-intercept form, we first need to rearrange them into the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.

First Equation:
[tex]\[ 5x = -2y + 5 \][/tex]

Let's start by isolating [tex]\( y \)[/tex]:

1. Add [tex]\( 2y \)[/tex] to both sides:
[tex]\[ 5x + 2y = 5 \][/tex]

2. Subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[ 2y = -5x + 5 \][/tex]

3. Finally, divide every term by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{5}{2}x + \frac{5}{2} \][/tex]

So, the first equation in slope-intercept form is:
[tex]\[ y = -\frac{5}{2}x + \frac{5}{2} \][/tex]

Second Equation:
[tex]\[ 20x = -8y + 20 \][/tex]

Similarly, we will isolate [tex]\( y \)[/tex]:

1. Add [tex]\( 8y \)[/tex] to both sides:
[tex]\[ 20x + 8y = 20 \][/tex]

2. Subtract [tex]\( 20x \)[/tex] from both sides:
[tex]\[ 8y = -20x + 20 \][/tex]

3. Divide every term by 8 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{20}{8}x + \frac{20}{8} \][/tex]

4. Simplify the fractions:
[tex]\[ y = -\frac{5}{2}x + \frac{5}{2} \][/tex]

So, the second equation in slope-intercept form is:
[tex]\[ y = -\frac{5}{2}x + \frac{5}{2} \][/tex]

Now, observe that both equations are:
[tex]\[ y = -\frac{5}{2}x + \frac{5}{2} \][/tex]

Since both equations represent the same line (they have the same slope and the same y-intercept), they do not intersect at a single point but rather overlap completely. Therefore, the system has an infinite number of solutions.