Problem 52 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. passes through \((3,7)\) and \((-2,-3)\)

Step-by-Step Solution

Answer

The equation is \(y = 2x + 1\).

Step 1: Find the slope of the line

The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated with the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((3,7)\) and \((-2,-3)\), we substitute into the formula: \(m = \frac{-3 - 7}{-2 - 3} = \frac{-10}{-5} = 2\). The slope is 2.

Step 2: Write the equation in point-slope form

Using the point-slope form of the equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is one of the points, substitute \((3,7)\) and \(m = 2\): \(y - 7 = 2(x - 3)\).

Step 3: Convert to slope-intercept form

To convert from point-slope form \(y - 7 = 2(x - 3)\) to slope-intercept form \(y = mx + b\), expand and simplify: \(y - 7 = 2x - 6\), then rearrange to \(y = 2x + 1\). The equation in slope-intercept form is \(y = 2x + 1\).

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is, basically rating how much the line goes up or down as it moves from left to right. In mathematical terms, the slope is often represented by the letter \(m\). It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. This can be found using the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of any two points on the line.
Understanding the concept of slope is crucial because it determines the direction of the line:

• If the slope is positive, the line goes upwards.
• If the slope is negative, the line descends.
• A zero slope means the line is horizontal.
• An undefined slope indicates a vertical line.

The case we examined found the slope \(m = 2\) by substituting the given points. This means that for every unit the line moves horizontally, it moves two units vertically.

Point-Slope Form

The point-slope form is a mathematical formula used to define the equation of a line when you know the slope and one point on the line. The formula is:

• \(y - y_1 = m(x - x_1)\)

Where \((x_1, y_1)\) is a known point and \(m\) is the slope.
This form is particularly useful for determining an equation quickly without needing the y-intercept right away.
In the worked exercise, we used the point \((3, 7)\) and the calculated slope \(m = 2\) to formulate:

• \(y - 7 = 2(x - 3)\)

This shows how effectively the point-slope form can capture a line's behavior with minimal information, enabling easy transition to other line forms like the slope-intercept form.

Equation of a Line

The equation of a line can be expressed in different forms, one of the most common being the slope-intercept form:

• \(y = mx + b\)

In this representation:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, where the line crosses the y-axis.

When we convert from point-slope to slope-intercept form, as done in the solution, we rearrange terms to solve for \(y\). This conversion aids in clearly identifying the slope and y-intercept. In our example, the line equation simplifies to:

• \(y = 2x + 1\)

This equation tells us the line rises by \(2\) units for each unit it moves to the right (thanks to the \(m = 2\)), and it crosses the y-axis at \(1\) (the \(b = 1\)).
The slope-intercept form is invaluable for graphing lines and quickly understanding their direction and position on a grid.