Sure, let's determine the slopes and y-intercepts for each of the given lines by converting each equation into the slope-intercept form [tex]\( y = mx + b \)[/tex].
### Line 1: [tex]\( 5y = 3x + 6 \)[/tex]
Step 1: Divide both sides by 5 to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3}{5}x + \frac{6}{5} \][/tex]
The slope ([tex]\( m \)[/tex]) of Line 1 is [tex]\(\frac{3}{5}\)[/tex] or 0.6, and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(\frac{6}{5}\)[/tex] or 1.2.
### Line 2: [tex]\( y = \frac{3}{5}x - 5 \)[/tex]
This equation is already in slope-intercept form.
The slope ([tex]\( m \)[/tex]) of Line 2 is [tex]\(\frac{3}{5}\)[/tex] or 0.6, and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(-5\)[/tex].
### Line 3: [tex]\( 10x - 6y = -8 \)[/tex]
Step 1: Rearrange the equation to isolate the [tex]\( y \)[/tex]-term.
[tex]\[ -6y = -10x - 8 \][/tex]
Step 2: Divide both sides by -6 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{10}{6}x + \frac{8}{6} \][/tex]
Step 3: Simplify the fractions:
[tex]\[ y = \frac{5}{3}x + \frac{4}{3} \][/tex]
The slope ([tex]\( m \)[/tex]) of Line 3 is [tex]\(\frac{5}{3}\)[/tex] or approximately 1.6667, and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(\frac{4}{3}\)[/tex] or approximately 1.3333.
### Summary:
- Line 1: [tex]\( y = \frac{3}{5}x + \frac{6}{5} \)[/tex], slope [tex]\( m_1 = 0.6 \)[/tex], y-intercept [tex]\( b_1 = 1.2 \)[/tex]
- Line 2: [tex]\( y = \frac{3}{5}x - 5 \)[/tex], slope [tex]\( m_2 = 0.6 \)[/tex], y-intercept [tex]\( b_2 = -5 \)[/tex]
- Line 3: [tex]\( y = \frac{5}{3}x + \frac{4}{3} \)[/tex], slope [tex]\( m_3 = 1.6667 \)[/tex], y-intercept [tex]\( b_3 = 1.3333 \)[/tex]
So, the slopes and y-intercepts for the lines are:
1. Line 1: [tex]\( m_1 = 0.6 \)[/tex], [tex]\( b_1 = 1.2 \)[/tex]
2. Line 2: [tex]\( m_2 = 0.6 \)[/tex], [tex]\( b_2 = -5 \)[/tex]
3. Line 3: [tex]\( m_3 = 1.6667 \)[/tex], [tex]\( b_3 = 1.3333 \)[/tex]
