Problem 77 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (5,-2) \text { and }(-4,7) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (5,-2) and (-4,7) is \(y = -x + 3\)

Step 1: Find the slope

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) and substitute the provided points into it. The slope \(m\) then becomes \(m = \frac{7 - (-2)}{-4 - 5} = -1\).

Step 2: Find the y-intercept

Substitute one of the points, say (5,-2), and slope \(m\) into the slope-intercept form \(y = mx + b\). So, -2 = -1*5 + b. After simplifying this equation, we get \(b = 3\).

Step 3: Write the Equation of the line

With the slope \(m = -1\) and the y-intercept \(b = 3\), the equation of the line that passes through both points in slope-intercept form is \(y = -x + 3\).

Key Concepts

Understanding the Slope Formula

The slope of a line is a measure of its steepness or inclination. It tells you how much the line rises or falls as you move from left to right on the graph. The slope formula is used to calculate this value when you know two points on the line. This formula is expressed as:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. The numerator \(y_2 - y_1\) represents the change in the y-coordinates, while the denominator \(x_2 - x_1\) represents the change in the x-coordinates.

For example, with the points (5,-2) and (-4,7), substitute these into the formula to get:

• \(m = \frac{7 - (-2)}{-4 - 5}\), which simplifies to \(m = -1\)

This slope of \(-1\) indicates the line falls by 1 unit as you move 1 unit to the right.

Decoding the Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. This point has an x-coordinate of zero and a y-coordinate equal to \(b\), which is derived from the line’s equation in slope-intercept form \(y = mx + b\).

To find the y-intercept when given a point and the slope, substitute these values into the formula. Using the point (5, -2) and the slope \(m = -1\):

• Substitute into \(y = mx + b\): \(-2 = -1 \times 5 + b\)
• Solve for \(b\): \(b = 3\)

This means the line crosses the y-axis at the point (0, 3). Ray tracing back to this concept, always remember:

• The y-intercept indicates the initial value of y when x is zero.

Crafting the Equation of a Line

Writing the equation of a line in slope-intercept form \(y = mx + b\) involves using the slope \(m\) and the y-intercept \(b\). This form provides a clear description of the line's behavior and position on the graph.

Returning to the specifics of our problem here, you discovered:

• slope \(m = -1\)
• y-intercept \(b = 3\)

With these values, the equation becomes:

• \(y = -x + 3\)

This depicts a line that decreases consistently, moving diagonally downwards to the right, with the initial y-value starting at 3 where the line intersects the y-axis.

Understanding how to write the equation of a line allows you to precisely represent linear relationships and easily predict y-values for given x-values.