Problem 30 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

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

Step 1: Calculate the Slope

Use the formula for finding the slope between two points, which is \( m = (y_2 - y_1) /(x_2 - x_1) \). Substituting in the given points, it becomes \( m = (-7 - 2) / (-7 - 2) = -1. \) The slope of the line, \( m \), is -1.

Step 2: Solve for the Y-Intercept

Now substitute one of the points and the slope into the equation \( y = mx + b \) to solve for the y-intercept, \( b \). You can choose either point; the result will be the same. If substituting (2,2) , your equation becomes \( 2 = -1*2 + b \). Simplifying, we get \( b = 2 - (-2) = 4 \).

Step 3: Write the Equation of the Line

With a slope, \( m \), of -1 and a y-intercept, \( b \), of 4, you can write the equation of the line as \( y = -x + 4 \). This is the equation of the line in slope-intercept form.

Key Concepts

Equation of a Line

When talking about the equation of a line, especially in algebra, one of the most common forms is the slope-intercept form. This form helps to easily understand and graph the line on a coordinate plane. The slope-intercept form of a line's equation is written as \( y = mx + b \). Here, \( m \) stands for the slope of the line, and \( b \) represents the y-intercept.
The equation expresses how the value of \( y \), which is the dependent variable, changes when \( x \), an independent variable, shifts. This relation reveals how steep the line is with the number \( m \), and where the line crosses the y-axis at the point \((0, b)\). Understanding this form allows us to quickly visualize how a line behaves, making it fundamental in solving and graphing linear equations.

Finding Slope

The slope of a line is a number that describes how steep a line is. It is the vertical change (rise) over the horizontal change (run) between any two points on the line. The formula for calculating this slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
To use this formula, you need two points through which the line passes, denoted as \((x_1, y_1)\) and \((x_2, y_2)\). For the given problem, these are \((2, 2)\) and \((-7, -7)\). By substituting these points into the formula, the slope comes out to be \(-1\).

• If the slope \( m \) is positive, the line rises from the left to the right.
• If it's negative, like in this case, the line falls as you move from left to right.
• A zero slope indicates a horizontal line, and an undefined slope marks a vertical line.

Grasping the concept of slope is essential for understanding how lines are oriented on the graph.

Y-Intercept Calculation

After determining the slope, the next step in the process is finding the y-intercept, which is the point where the line crosses the y-axis. This is essential for constructing the line's equation in slope-intercept form. Using the equation \( y = mx + b \), we need to solve for \( b \).
Substitute one of the line's points and the calculated slope into the equation. For instance, using the point \((2, 2)\), apply the values: \( 2 = -1(2) + b \). Solving for \( b \) gives \( b = 4 \). Therefore, the y-intercept is \( 4 \), indicating the line intersects the y-axis at \((0, 4)\).
This calculation is vital because it finalizes the full equation of the line as \( y = -x + 4 \), ensuring that both the slope and y-intercept accurately represent how the line behaves and where exactly it is located on the graph.