Problem 56

Question

Find an equation of the line that goes through the given point and has the given slope. Give the answer in slope-intercept form. See Example 5 (-3, -1) with slope 6

Step-by-Step Solution

Verified

Answer

The equation is $ y = 6x + 17 $.

1Step 1: Identify the point and slope

The point given is (-3, -1) and the slope is 6.

2Step 2: Understand the slope-intercept form

The slope-intercept form of a line is given by the equation: $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.

3Step 3: Substitute the slope and point into the slope-intercept form

Use the given slope (6) and the coordinates of the given point (-3, -1) to write the equation: Substitute $ m = 6 $, $ x = -3 $, and $ y = -1 $ into the equation $ y = mx + b $: -1 = 6(-3) + b

4Step 4: Solve for the y-intercept $ b $

Calculate $ 6(-3) $: -1 = -18 + b Add 18 to both sides to isolate $ b $: -1 + 18 = b So, $ b = 17 $

5Step 5: Write the final equation

Substitute $ m = 6 $ and $ b = 17 $ back into the slope-intercept form equation: $ y = 6x + 17 $

Key Concepts

Linear EquationsCalculating y-interceptSlope and Points

Linear Equations

Linear equations describe straight lines on a graph. The standard equation for a straight line is represented in the slope-intercept form: \[ y = mx + b \]. In this equation, $ m $ stands for the slope of the line, and $ b $ stands for the y-intercept, the point where the line crosses the y-axis.Understanding linear equations makes it easier to plot relationships between variables on a graph. These are foundational in algebra and come in handy when solving real-world problems involving constant rates of change or linear relationships.

Calculating y-intercept

The y-intercept ($ b $) is the value of $ y $ where the line intersects the y-axis. This happens when $ x = 0 $. Here's how you can find it:

Start with the slope-intercept form equation: $ y = mx + b $.
Substitute a given point on the line ($ x, y $) and the slope ($ m $) into the equation.
Solve for $ b $ by isolating it on one side of the equation.

For instance, if you are given a point (-3, -1) and a slope of 6:

Substitute into the formula: $ -1 = 6(-3) + b $.
Simplify the equation:$ -1 = -18 + b $.
Solve for $ b $: $ -1 + 18 = b $, so $ b = 17 $.

The y-intercept, $ b $, is 17.

Slope and Points

The slope ($ m $) measures the steepness of a line and is calculated as the ratio of the 'rise' (vertical change) over the 'run' (horizontal change) between two points on the line. If you have a point ($ x_1, y_1 $) and the given slope $ m $, you can easily set up your linear equation:

Start with the formula: $ y = mx + b $.
Input the values for $ x $ and $ y $ from the point into the formula along with $ m $.

Using the given point (-3, -1) and slope (6), you can write it as: $ -1 = 6(-3) + b $.

Solve for $ b $ as explained in the previous section.
Finally, substitute $ m $ and $ b $ back into the slope-intercept form: $ y = 6x + 17 $.

This gives you the complete equation of the line passing through the given point with the specified slope.

Problem 55

Problem 56

Other exercises in this chapter

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