Chapter4: Systems Of Linear Equations
Section4.1: Solve Systems Of Linear Equations With Two Variables
Problem 68E: In a system of linear equations, the two equations have the same intercepts. Describe the possible...
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
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell