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Chapter 1: Problem 8

If possible, write each equation in the form \(y=m x+b .\) Then identify the slope and the \(y\) -intercept. $$y=\frac{1}{5}(10 x+5)-5+7 x$$

Short Answer

Expert verified
Slope: 9, y-Intercept: -4

Step by step solution

01

Distribute \frac{1}{5} Through the Parentheses

To start, distribute \frac{1}{5} through the parentheses in the equation. This means you multiply \frac{1}{5} by each term inside the parentheses: $$ y = \frac{1}{5}(10x) + \frac{1}{5}(5) - 5 + 7x$$
02

Simplify the Distributed Terms

Simplify each term that was distributed: $$ y = 2x + 1 - 5 + 7x $$
03

Combine Like Terms

Combine the like terms (terms with x and constant terms): $$ y = (2x + 7x) + (1 - 5) $$ which simplifies to $$ y = 9x - 4 $$
04

Identify the Slope (m) and y-Intercept (b)

Now the equation is in the form y = mx + b, where m is the slope and b is the y-intercept. From the equation $$ y = 9x - 4 $$, identify the slope and the y-intercept. The slope (m) is 9, and the y-intercept (b) is -4.

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Key Concepts

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

slope
A slope tells us how steep a line is on a graph. It's the 'm' in the equation of a line: y = mx + b.
The slope measures the rise over run, or how much y changes for each change in x.
In simpler terms, it's how much the line goes up or down when you move to the right.
In the equation y = 9x - 4, the slope (m) is 9. This means that for every 1 unit you move to the right on the x-axis, the line goes up by 9 units on the y-axis.

A positive slope means the line rises as it goes right, while a negative slope means the line falls as it goes right.
The higher the absolute value of the slope, the steeper the line.
y-intercept
The y-intercept is where the line crosses the y-axis. It's the 'b' in the equation of a line: y = mx + b.
This is the y-value when x is 0.
In the equation y = 9x - 4, the y-intercept (b) is -4. This means the line crosses the y-axis at (0, -4).

The y-intercept gives us a starting point for graphing the line. From there, you can use the slope to find other points.
If the y-intercept is positive, the line crosses above the origin; if negative, it crosses below.
combining like terms
Combining like terms simplifies an equation by merging terms with the same variable.
In the step-by-step solution, we combined 2x and 7x to get 9x.
We also combined the constant terms 1 and -5 to get -4.

Combining like terms helps to bring an equation to its simplest form, making it easier to solve.
Always group all x-terms together and all constant terms together.
This process transforms complex equations into simpler, more manageable forms.
distributive property
The distributive property helps us remove parentheses by distributing a multiplication over an addition or subtraction.
Symbolically, this property states that a(b + c) = ab + ac.
In our example, we used \( \frac{1}{5} (10x + 5) = \frac{1}{5} * 10x + \frac{1}{5} * 5 \) which simplifies to 2x + 1.

This method helps break down complex expressions into simpler ones.
It's particularly useful for solving linear equations and making them easier to understand.
It ensures that every term inside the parentheses is multiplied by the factor outside the parentheses.

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Most popular questions from this chapter

Carlos and Shondra were designing posters for the school play. During the first two days, they created 40 posters. By the third day, they had established a routine, and they calculated that together they would produce 20 posters an hour. a. Make a table like the following that shows how many posters Carlos and Shondra make as they work through the third day. b. Draw a graph to represent the number of posters Carlos and Shondra will make as they work through the third day. c. Write an equation to represent the number of posters they will make as they work through the third day. d. Make a table to show the total number of posters they will have as they work through the third day. e. Draw a graph to show the total number of posters Carlos and Shondra will have as they work through the third day. f. Write an equation that will allow you to calculate the total number of posters they will have based on the number of hours they work. g. Explain how describing just the number of posters created the third day is different from describing the total number of posters created. Is direct variation involved? How are these differences represented in the tables, the graphs, and the equations?

For each equation, identify the slope and the y-intercept. Graph the line to check your answer. $$ y=x-3 $$

Consider this data set. $$\begin{array}{|c|c|c|c|c|c|}\hline x & {0} & {2} & {4} & {6} & {8} \\\ \hline y & {2} & {20} & {6} & {8} & {10} \\ \hline\end{array}$$ a. Graph the data set. b. One point is an outlier. Which point is it? c. Find the mean of the x values and the mean of the y values. d. Try to find a line that is a good fit for the data and goes through the point (mean of \(x\) values, mean of \(y\) values). Write an equation for your line. e. Now find the means of the variables, ignoring the outlier. In other words, do not include the values for the outlier in your calculations. f. Try to find a new line that is a good fit for the data, using the means you calculated in Part e for the (mean of x values, mean of y values) point. Write an equation for your line. g. Do you think either line should be considered the best fit for the data? Explain.

Just as the \(y\) -intercept of a line is the \(y\) value at which the line crosses the \(y\) -axis, the \(x\) -intercept is the \(x\) value at which the line crosses the \(x\) -axis. In Exercises \(22-25,\) find an equation of a line with the given \(x\) -intercept and slope. \(x\) -intercept \(-2,\) slope \(-\frac{1}{2}\)

For each set of equations, tell what the graphs of all four relationships have in common without drawing the graphs. Explain your answers. $$\begin{array}{l}{y=1-x} \\ {y=1-2 x} \\ {y=1-3 x} \\ {y=1-4 x}\end{array}$$
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