Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Solving Systems of Equations by Graphing
#### Problem:
Solve the system of equations by graphing:
\[
\begin{cases}
y = -2x - 1 \\
2x + 4y = 8
\end{cases}
\]
#### Solution:
1. **Graph the First Equation**: \(y = -2x - 1\)
- This is a linear equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- The y-intercept \(b = -1\), which means the line crosses the y-axis at \((0, -1)\).
- The slope \(m = -2\) indicates that for every 1 unit increase in \(x\), \(y\) decreases by 2 units.
2. **Graph the Second Equation**: \(2x + 4y = 8\)
- First, simplify this equation to the slope-intercept form \(y = mx + b\).
- \(2x + 4y = 8\)
- Subtract \(2x\) from both sides: \(4y = -2x + 8\).
- Divide by 4: \(y = -\frac{1}{2}x + 2\).
- This line crosses the y-axis at \((0, 2)\) and has a slope of \(-\frac{1}{2}\).
#### Instructions for Graphing:
1. **Plot the first line \(y = -2x - 1\)**:
- Start at the point \((0, -1)\) on the y-axis.
- From there, use the slope \(-2\) to find another point. For instance, if \(x = 1\), then \(y = -2(1) - 1 = -3\). So, plot the point \((1, -3)\).
- Draw a straight line through these points.
2. **Plot the second line \(y = -\frac{1}{2}x + 2\)**:
- Start at the point \((0, 2)\) on the y-axis.
- From there, use the slope \(-\frac{1}{2}\) to find another point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa369931a-fe5c-47f4-90de-c0f8ffd54c08%2F67997bce-395a-4996-9aff-bc726c9659e9%2Fcdvuite_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Systems of Equations by Graphing
#### Problem:
Solve the system of equations by graphing:
\[
\begin{cases}
y = -2x - 1 \\
2x + 4y = 8
\end{cases}
\]
#### Solution:
1. **Graph the First Equation**: \(y = -2x - 1\)
- This is a linear equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- The y-intercept \(b = -1\), which means the line crosses the y-axis at \((0, -1)\).
- The slope \(m = -2\) indicates that for every 1 unit increase in \(x\), \(y\) decreases by 2 units.
2. **Graph the Second Equation**: \(2x + 4y = 8\)
- First, simplify this equation to the slope-intercept form \(y = mx + b\).
- \(2x + 4y = 8\)
- Subtract \(2x\) from both sides: \(4y = -2x + 8\).
- Divide by 4: \(y = -\frac{1}{2}x + 2\).
- This line crosses the y-axis at \((0, 2)\) and has a slope of \(-\frac{1}{2}\).
#### Instructions for Graphing:
1. **Plot the first line \(y = -2x - 1\)**:
- Start at the point \((0, -1)\) on the y-axis.
- From there, use the slope \(-2\) to find another point. For instance, if \(x = 1\), then \(y = -2(1) - 1 = -3\). So, plot the point \((1, -3)\).
- Draw a straight line through these points.
2. **Plot the second line \(y = -\frac{1}{2}x + 2\)**:
- Start at the point \((0, 2)\) on the y-axis.
- From there, use the slope \(-\frac{1}{2}\) to find another point.
![**Graphing Inequalities: Example**
Consider the following inequalities:
\[
\begin{cases}
y > x - 1 \\
y < -4x + 1
\end{cases}
\]
### Explanation of the Graph
In the graph, the system of linear inequalities is represented on a Cartesian plane. Here, each point on the plane corresponds to a pair of \( (x, y) \) values.
#### Axes:
- The x-axis (horizontal) ranges from -5 to 5.
- The y-axis (vertical) ranges from -5 to 5.
#### Equations:
1. \( y > x - 1 \)
2. \( y < -4x + 1 \)
### How to Interpret the Graph
1. **Graphing \( y > x - 1 \):**
- This inequality represents the region above the line \( y = x - 1 \).
- To identify the line \( y = x - 1 \), note the slope is 1 and the y-intercept is -1.
- The region above this line (not including the line itself) satisfies the inequality.
2. **Graphing \( y < -4x + 1 \):**
- This inequality represents the region below the line \( y = -4x + 1 \).
- To identify the line \( y = -4x + 1 \), note the slope is -4 and the y-intercept is 1.
- The region below this line (not including the line itself) satisfies the inequality.
### Analyzing the Graph:
The two inequalities intersect, and their combined regions define a specific area in the Cartesian plane.
- Region for \( y > x - 1 \): Above the line \( y = x - 1 \).
- Region for \( y < -4x + 1 \): Below the line \( y = -4x + 1 \).
We intersect these regions to find the area that satisfies both conditions simultaneously. This particular area will lie between these lines where both conditions are met.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa369931a-fe5c-47f4-90de-c0f8ffd54c08%2F67997bce-395a-4996-9aff-bc726c9659e9%2Fhaujt2i_processed.png&w=3840&q=75)
Transcribed Image Text:**Graphing Inequalities: Example**
Consider the following inequalities:
\[
\begin{cases}
y > x - 1 \\
y < -4x + 1
\end{cases}
\]
### Explanation of the Graph
In the graph, the system of linear inequalities is represented on a Cartesian plane. Here, each point on the plane corresponds to a pair of \( (x, y) \) values.
#### Axes:
- The x-axis (horizontal) ranges from -5 to 5.
- The y-axis (vertical) ranges from -5 to 5.
#### Equations:
1. \( y > x - 1 \)
2. \( y < -4x + 1 \)
### How to Interpret the Graph
1. **Graphing \( y > x - 1 \):**
- This inequality represents the region above the line \( y = x - 1 \).
- To identify the line \( y = x - 1 \), note the slope is 1 and the y-intercept is -1.
- The region above this line (not including the line itself) satisfies the inequality.
2. **Graphing \( y < -4x + 1 \):**
- This inequality represents the region below the line \( y = -4x + 1 \).
- To identify the line \( y = -4x + 1 \), note the slope is -4 and the y-intercept is 1.
- The region below this line (not including the line itself) satisfies the inequality.
### Analyzing the Graph:
The two inequalities intersect, and their combined regions define a specific area in the Cartesian plane.
- Region for \( y > x - 1 \): Above the line \( y = x - 1 \).
- Region for \( y < -4x + 1 \): Below the line \( y = -4x + 1 \).
We intersect these regions to find the area that satisfies both conditions simultaneously. This particular area will lie between these lines where both conditions are met.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 6 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education