4.3 Q-24

Question Statement

Find the equation of a line through the point (-4, 7) that is parallel to the line given by the equation: $2x - 7y + 4 = 0$

Background and Explanation

To solve this problem, we need to recall that two lines are parallel if and only if they have the same slope. The given line, $2x - 7y + 4 = 0$ , has a specific slope, and we will use this slope to find the equation of the line through (-4, 7). We will use the point-slope form of the equation of a line to derive the required equation.

Solution

Step 1: Find the slope of the given line.

We start by rewriting the equation of the given line in slope-intercept form, $y = mx + b$ , where $m$ is the slope.

Starting from the equation: $2x - 7y + 4 = 0$

Isolate $y$ : $2x + 4 = 7y$

Divide by 7: $y = \frac{2}{7}x + \frac{4}{7}$

So, the slope ( $m$ ) of the given line is: $m = \frac{2}{7}$

Step 2: Use the point-slope form of the equation of a line.

Since parallel lines have the same slope, the slope of our required line is also $\frac{2}{7}$ . Now, we can apply the point-slope form of the equation: $y - y_1 = m(x - x_1)$

Here, $m = \frac{2}{7}$ , and the point through which the line passes is (-4, 7), so $x_1 = -4$ and $y_1 = 7$ .

Substitute these values into the point-slope form: $y - 7 = \frac{2}{7}(x - (-4))$

Step 3: Simplify the equation.

Simplifying the right-hand side: $y - 7 = \frac{2}{7}(x + 4)$

Multiply both sides by 7 to eliminate the denominator: $7(y - 7) = 2(x + 4)$

Distribute both sides: $7y - 49 = 2x + 8$

Now, rearrange the equation: $2x - 7y + 57 = 0$

Thus, the equation of the required line is: $2x - 7y + 57 = 0$

Key Formulas or Methods Used

Point-Slope Form: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
Slope of a Line: The slope of the given line was found by converting the equation to slope-intercept form $y = mx + b$ .

Summary of Steps

Find the slope of the given line by converting its equation to slope-intercept form.
Use the point-slope form of the equation of a line to set up the equation.
Substitute the given point (-4, 7) and the slope $\frac{2}{7}$ .
Simplify the equation to obtain the final answer: $2x - 7y + 57 = 0$ .