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.
