To find the equation of the line that passes through the point [tex]\((3,7)\)[/tex] and is parallel to the line given by the equation [tex]\(5x - 8y = 4\)[/tex], we can follow these steps:
### Step 1: Determine the slope of the given line.
First, we need to find the slope of the line [tex]\(5x - 8y = 4\)[/tex]. We can rewrite this equation in the slope-intercept form [tex]\(y = mx + b\)[/tex].
Start by isolating [tex]\(y\)[/tex]:
[tex]\[ 5x - 8y = 4 \][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[ -8y = -5x + 4 \][/tex]
Divide every term by [tex]\(-8\)[/tex] to solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{5}{8}x - \frac{4}{8} \][/tex]
Simplify:
[tex]\[ y = \frac{5}{8}x - \frac{1}{2} \][/tex]
The slope ([tex]\(m\)[/tex]) of this line is [tex]\(\frac{5}{8}\)[/tex].
### Step 2: Since the new line is parallel to the given line, it has the same slope.
The slope of our new line is also [tex]\(\frac{5}{8}\)[/tex].
### Step 3: Use the point-slope form to find the equation of the new line.
The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where [tex]\((x_1, y_1)\)[/tex] is the point the line passes through and [tex]\(m\)[/tex] is the slope. Substituting [tex]\(m = \frac{5}{8}\)[/tex] and the point [tex]\((3, 7)\)[/tex]:
[tex]\[ y - 7 = \frac{5}{8}(x - 3) \][/tex]
### Step 4: Simplify the equation to the slope-intercept form [tex]\(y = mx + b\)[/tex].
Distribute the slope [tex]\(\frac{5}{8}\)[/tex] on the RHS:
[tex]\[ y - 7 = \frac{5}{8}x - \frac{15}{8} \][/tex]
Add 7 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{5}{8}x - \frac{15}{8} + 7 \][/tex]
Convert 7 to a fraction with a common denominator:
[tex]\[ y = \frac{5}{8}x - \frac{15}{8} + \frac{56}{8} \][/tex]
Combine the fractions:
[tex]\[ y = \frac{5}{8}x + \frac{41}{8} \][/tex]
### Step 5: Write the final equation.
The equation of the line that passes through [tex]\((3,7)\)[/tex] and is parallel to [tex]\(5x - 8y = 4\)[/tex] is:
[tex]\[ y = \frac{5}{8}x + \frac{41}{8} \][/tex]
To summarize:
- The slope of the parallel line is [tex]\(\frac{5}{8}\)[/tex].
- The y-intercept of the new line is [tex]\(\frac{41}{8}\)[/tex].
- The equation of the line is [tex]\(y = \frac{5}{8}x + \frac{41}{8}\)[/tex].
