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Chapter 1: Problem 9

If possible, write each equation in the form \(y=m x+b .\) Then identify the slope and the \(y\) -intercept. $$3 x+2(x+1)=-\frac{1}{2}(4 x+6)+y$$

Short Answer

Expert verified
Slope: 7, y-intercept: 5

Step by step solution

01

- Simplify the left side

Combine the terms on the left side: \[ 3x + 2(x+1) \]Distribute the 2: \[ 3x + 2x + 2 = 5x + 2 \]
02

- Simplify the right side

Distribute the \frac{1}{2}: \[ -\frac{1}{2}(4x+6) = -2x - 3 \]So the equation becomes: \[ 5x + 2 = -2x - 3 + y \]
03

- Solve for y

Move all terms except y to the left side: \[ 5x + 2 + 2x + 3 = y \]Simplify: \[ 7x + 5 = y \]Now the equation is in the form \[ y = mx + b \]
04

- Identify the slope and y-intercept

In the form \[ y = mx + b \]\( m \) is the slope and \( b \) is the y-intercept. From \[ y = 7x + 5 \], the slope \( m = 7 \) and the y-intercept \( b = 5 \)

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Key Concepts

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

linear equations
Linear equations are essential in algebra and can describe a straight line on a graph. They generally take the form \[ y = mx + b \], where:
  • \( y \) represents the dependent variable or the output of the function.
  • \( m \) is the slope of the line, showing how steep the line is.
  • \( x \) is the independent variable or the input of the function.
  • \( b \) is the \( y \)-intercept, the point where the line crosses the \( y \)-axis.
In this problem, we start with a complex equation and simplify it step by step into the slope-intercept form. This form (\( y = mx + b \)) is straightforward to understand and quickly gives us the slope and \( y \)-intercept. Linear equations are foundational because they help us understand relationships between variables, allowing us to make predictions and analyze trends.
slope
The slope (\( m \)) of a line is an indication of its steepness and direction. It shows how much \( y \) changes for a unit change in \( x \). In the slope-intercept form (\( y = mx + b \)), \( m \) directly tells us this rate of change.
  • If the slope \( m \) is positive, the line rises as it moves from left to right.
  • If \( m \) is negative, the line falls as it moves from left to right.
  • A larger absolute value of \( m \) means a steeper slope.
In the given problem, after simplifying, we found that the slope \( m \) is 7. This means for every unit increase in \( x \), \( y \) increases by 7 units. Understanding slope is key in graphing linear equations, as it helps us draw the line accurately and understand the relationship between the variables.
y-intercept
The \( y \)-intercept (\( b \)) is the point where the line crosses the \( y \)-axis. This tells us the value of \( y \) when \( x \) is zero.
In the slope-intercept form \( y = mx + b \), \( b \) is easily identified and provides useful information about the starting point of the relationship between \( x \) and \( y \).
  • For example, if \( b \) is 5, the line crosses the \( y \)-axis at (0, 5).
In the given exercise, after simplifying the equation, we found that the \( y \)-intercept \( b \) is 5. This means the line crosses the \( y \)-axis at the point (0, 5). The \( y \)-intercept is crucial for graphing linear equations, as it gives us a starting point to draw the line correctly.
algebra
Algebra involves manipulating symbols and numbers to solve equations. Understanding how to use algebra to simplify and solve equations is key to mastering math.
  • In our example, we started with a complex equation and used algebraic steps to simplify it.
  • We combined like terms, distributed coefficients, and moved terms across the equation to isolate \( y \).
This process required a good grasp of algebraic principles, such as the distributive property and combining like terms. Finally, we arrived at the slope-intercept form \( y = mx + b \), making it easy to identify the slope and \( y \)-intercept. Algebra is a powerful tool that helps us solve real-world problems by creating and simplifying equations.

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Most popular questions from this chapter

For each equation, identify the slope and the y-intercept. Graph the line to check your answer. $$ y=2 x+0.25 $$

Evaluate each expression for \(a=2\) and \(b=3\). $$ \left(\frac{a}{b}+a\right)^{a} $$

The table shows x and y values for a particular relationship. $$\begin{array}{|c|c|c|c|c|}\hline x & {6} & {3} & {1} & {2.5} \\ \hline y & {7} & {1} & {-3} & {0} \\ \hline\end{array}$$ $$\begin{array}{l}{\text { a. Graph the ordered pairs }(x, y) . \text { Make each axis scale from }-10} \\ {\text { to } 10 .} \\ {\text { b. Could the points represent a linear relationship? If so, write an }} \\ {\text { equation for the line. }} \\ {\text { c. From your graph, predict the } y \text { value for an } x \text { value of }-2 . \text { Check }} \\ {\text { your answer by substituting it into the equation. }}\end{array}$$ $$ \begin{array}{l}{\text { d. From your graph, find the } x \text { value for a } y \text { value of }-2 . \text { Check your }} \\ {\text { answer by substituting it into the equation. }} \\ {\text { e. Use your equation to find the } y \text { value for each of these } x \text { values: }} \\ {0,-- 1,-1.5,-2.5 . \text { Check that the corresponding points all lie on }} \\\ {\text { the line. }}\end{array} $$

For Exercises 20鈥�28, answer Parts a and b. a. What is the constant difference between the \(y\) values as the \(x\) values increase by 1\(?\) b. What is the constant difference between the \(y\) values as the \(x\) values decrease by 2\(?\) $$ y=23 x-18 $$

Alejandro looked at the equations \(y=\frac{3}{2} x-1\) and \(y=-\frac{2}{3} x+2\) and said, These lines form a right angle. a. Graph both lines on two different grids, with the axes labeled as shown here. b. Compare the lines on each grid. Do both pairs of lines form a right angle? c. What kind of assumption must Alejandro have made when he said the lines form a right angle?
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