Problem 7 Solve each system graphically. ... [FREE SOLUTION]

Chapter 15: Problem 7

Solve each system graphically. $$ \begin{aligned} &4 x-20=5 y\\\ &8 x-10 y=12 \end{aligned} $$

Short Answer

Expert verified

No solution; the lines are parallel.

Step by step solution

- Rewrite the equations in slope-intercept form

Start by rewriting each equation in the slope-intercept form, which is of the form y = mx + b.For the first equation, $ 4x - 20 = 5y $:1. Add 20 to both sides: $ 4x = 5y + 20 $.2. Subtract 5y from both sides: $ 4x - 20 = 5y $.3. We then divide both sides by 5: $ y = \frac{4}{5}x - 4 $.For the second equation, $ 8x - 10y = 12 $:1. Isolate -10y by subtracting 8x from both sides: $ -10y = -8x + 12 $.2. Divide both sides by -10: $ y = \frac{8}{10}x - \frac{12}{10} $.3. Simplify the fraction: $ y = \frac{4}{5}x - 1.2 $.

- Graph both equations

Using graph paper or an online graphing tool, plot the two lines represented by the equations $ y = \frac{4}{5}x - 4 $ and $ y = \frac{4}{5}x - 1.2 $.For the first line, $ y = \frac{4}{5}x - 4 $:1. Start at the y-intercept (0, -4).2. Use the slope ($ \frac{4}{5} $) to plot another point by moving 4 units up and 5 units right.3. Draw the line through these points.For the second line, $ y = \frac{4}{5}x - 1.2 $:1. Start at the y-intercept (0, -1.2).2. Use the same slope ($ \frac{4}{5} $) to plot another point by moving 4 units up and 5 units right.3. Draw the line through these points.

- Determine the point of intersection

Observe the graph to determine where the two lines intersect. Since both lines have the same slope but different y-intercepts, the lines are parallel and do not intersect.

- Conclusion

Since the lines do not intersect, the system of equations has no solution.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Graphing Equations

When solving a system of equations graphically, we start by plotting each line on a graph. This helps us see where the lines intersect, which represents the solution to the system.
To graph these equations, we need to follow these steps:

Rewrite each equation in slope-intercept form (y = mx + b).
Plot the y-intercept, the point where the line crosses the y-axis.
Use the slope to determine another point on the line.

By connecting these points, you can draw the line for each equation. The intersection of the lines is the solution. However, if the lines are parallel, there is no intersection and hence no solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as y = mx + b, where:

'm' is the slope of the line.
'b' is the y-intercept, the point where the line crosses the y-axis.

To convert from standard form (Ax + By = C) to slope-intercept form:

Isolate y on one side.
Move the other terms to the opposite side.
Divide by the coefficient of y, if necessary.

For example, given 4x - 20 = 5y, we rearrange it to y = (4/5)x - 4. This form makes it easy to graph the equation and understand the line's characteristics.

Parallel Lines

Parallel lines have the same slope but different y-intercepts. This means they will never meet; they run alongside each other forever. When graphing two linear equations, if both lines have the same slope and different intercepts, they are parallel.
In the exercise:

Both lines are y = (4/5)x - 4 and y = (4/5)x - 1.2.

Since the slopes (4/5) are identical but the y-intercepts (-4 and -1.2) are different, the lines are parallel. This parallel nature indicates that the system of equations has no solution because the lines do not intersect.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve each system graphically. $$ \begin{aligned} &4 x-20=5 y\\\ &8 x-10 y=12 \end{aligned} $$

Short Answer

Step by step solution

- Rewrite the equations in slope-intercept form

- Graph both equations

- Determine the point of intersection

- Conclusion

Key Concepts

Graphing Equations

Slope-Intercept Form

Parallel Lines

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Discrete Mathematics

Decision Maths

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.