Problem 6 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(-12,2), m=-2$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -2x -22\).

Step 1: Substitute given values into slope-intercept form

With the given point (-12, 2) and slope -2, we substitute these values into the slope-intercept form \(y = mx + b\). We get 2 = -2 * -12 + b.

Step 2: Solve for b

By simplifying the equation, we find that 2 = 24 + b. Solving for \(b\) yields \(b\) = 2 - 24 = -22.

Step 3: Write final equation

With \(m\) = -2 and \(b\) = -22, the final line equation in slope-intercept form is \(y = -2x -22\).

Key Concepts

Linear Equations

Linear equations are fundamental to algebra and describe a straight line on a graph. They are usually written in the form of the equation:

• Standard form: \(ax + by = c\)
• Slope-intercept form: \(y = mx + b\)

Both forms represent linear relationships where the variables correspond to coordinates on a Cartesian plane.
The slope-intercept form is especially useful because it provides immediate insight into two critical aspects of the line:

• The slope (m) which determines the steepness or direction.
• The y-intercept (b) which indicates where the line crosses the y-axis.

It's important when given a point and slope to use this form to easily determine the equation. This form allows for straightforward substitution and solving.

Slope

The slope of a line is a measure of its steepness, representing the ratio of vertical change (rise) to horizontal change (run) between two points on the line. In the slope-intercept formula \(y = mx + b\), \(m\) stands for the slope:

• Positive slope: Line rises as it moves from left to right.
• Negative slope: Line falls as it moves from left to right.
• Zero slope: Line is horizontal; no rise.
• Undefined slope: Line is vertical; no run.

Understanding slope is crucial in determining how a line behaves.
In the original exercise, a slope of \(-2\) reveals that for every unit increase in \(x\), the value of \(y\) decreases by 2 units. This negative slope indicates a line that descends from left to right on the graph.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis of a graph. In a linear equation written in slope-intercept form, \(y = mx + b\), the intercept is denoted by \(b\). This value shows where the line intersects the y-axis when the value of \(x\) is zero.

• It determines the starting point of the line on the y-axis.
• It provides a specific location on the graph that helps anchor the line for further plotting using the slope.

From the solution of the problem presented, once the slope was applied at the point \((-12, 2)\), solving the equation revealed the y-intercept: \(-22\).
This means when \(x\) is zero, the line crosses the y-axis at -22, which completes the plotting of the line on a graph.