Problem 59 Write an equation for each line ... [FREE SOLUTION]

Chapter 3: Problem 59

Write an equation for each line passing through the given point and having the given slope. Give the final answer in slope-intercept form. See Examples 5 and $6 .$ $$ (-2,5), m=\frac{2}{3} $$

Short Answer

Expert verified

The equation of the line is $ y = \frac{2}{3}x + \frac{19}{3} $.

Step by step solution

- Understand the slope-intercept form

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

- Substitute the slope and point into the slope-intercept form

We know the slope $ m = \frac{2}{3} $ and the point $ (-2, 5) $. Substitute these into the equation. First, write the equation with the slope: $ y = \frac{2}{3}x + b $.

- Substitute the point to find the y-intercept

Substitute $ x = -2 $ and $ y = 5 $ into the equation to find $ b $. $ 5 = \frac{2}{3}(-2) + b $ Simplify and solve for $ b $: $ 5 = -\frac{4}{3} + b $.

- Solve for the y-intercept

Add $ \frac{4}{3} $ to both sides to isolate $ b $: $ 5 + \frac{4}{3} = b $. Convert $ 5 $ to a fraction: $ \frac{15}{3} + \frac{4}{3} = b $. Simplify: $ \frac{19}{3} = b $.

- Write the final equation

Now that we have the slope $ m = \frac{2}{3} $ and the y-intercept $ b = \frac{19}{3} $, substitute them back into the slope-intercept form equation: $ y = \frac{2}{3}x + \frac{19}{3} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations

Understanding linear equations is core to tackling many algebra problems. A linear equation is a mathematical expression that creates a straight line when graphed on a coordinate plane. The general form of a linear equation is: \[ y = mx + b \] Here,

$ y $: represents the dependent variable (usually the y-coordinate).
$ x $: represents the independent variable (usually the x-coordinate).
$ m $: represents the slope of the line, which shows the rate of change.
$ b $: represents the y-intercept, which is the point where the line crosses the y-axis.

Linear equations appear in various real-world scenarios, from calculating distances to predicting profits. The key is always to understand the configuration and the significance of the slope and y-intercept. Grasping these concepts can simplify complex problems and make algebra more intuitive.

Slope

The slope of a line is a measure of its steepness and direction. Mathematically, it's defined as: \[ m = \frac{\Delta y}{\Delta x} \] This means the slope$ m $ is calculated by dividing the change in y (vertical difference) by the change in x (horizontal difference). The slope tells us how much y increases (or decreases) as x increases.

A positive slope means the line inclines upward as it moves from left to right.
A negative slope means the line declines downward.
A zero slope means the line is horizontal.
An undefined slope means the line is vertical.

In our problem, the given slope is $ \frac{2}{3} $, meaning that for every 3 units x increases, y increases by 2 units.

Y-Intercept

The y-intercept of a line is where it crosses the y-axis. In the slope-intercept form \[ y = mx + b \]$ b $ is the y-intercept. Determining the y-intercept is straightforward once you have the slope and a point on the line. You can substitute the values into the equation to solve for $ b $. In our problem, we started with the slope $ \frac{2}{3} $ and a point (-2, 5). We substituted these values into the equation to find the y-intercept:

Step 1: Plug in the point and the slope: $ 5 = \frac{2}{3}(-2) + b $
Step 2: Simplify: $ 5 = -\frac{4}{3} + b $
Step 3: Solve for $ b $:$ b = \frac{19}{3} $

This final result gives us the complete equation in slope-intercept form.

Substitution Method

The substitution method is a powerful tool for solving linear equations, especially when working with slope-intercept form. Substitution involves replacing one variable with its actual value to find the unknown terms. In our problem, we knew the slope $ m = \frac{2}{3} $ and the point (-2, 5). Here鈥檚 how we used substitution to find the y-intercept:

Start with the slope-intercept form: $ y = \frac{2}{3}x + b $
Substitute the known values (x = -2, y = 5): $ 5 = \frac{2}{3}(-2) + b $
Simplify: $ 5 = -\frac{4}{3} + b $
Isolate $ b $: $ 5 + \frac{4}{3} = b $
Convert 5 to a fraction: $ \frac{15}{3} + \frac{4}{3} = b $
Solve $ b $: $ b = \frac{19}{3} $

This method is efficient and helps you find the missing components of the equation to express it fully in the slope-intercept form: \[ y = \frac{2}{3}x + \frac{19}{3} \]. Learning and practicing this method can make solving linear equations much easier.

91影视

Write an equation for each line passing through the given point and having the given slope. Give the final answer in slope-intercept form. See Examples 5 and \(6 .\) $$ (-2,5), m=\frac{2}{3} $$

Short Answer

Step by step solution

- Understand the slope-intercept form

- Substitute the slope and point into the slope-intercept form

- Substitute the point to find the y-intercept

- Solve for the y-intercept

- Write the final equation

Key Concepts

Linear Equations

Slope

Y-Intercept

Substitution Method

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