/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Use the graphical method to solv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 18

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{l}8 x-6 y=24 \\ 4 x-3 y=12\end{array}\right.$$

Short Answer

Expert verified
The system has infinitely many solutions along the line \( y = \frac{4}{3} x - 4 \).

Step by step solution

01

Rewrite equations in slope-intercept form

Convert each equation into the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.ewlineFor the first equation \(8x - 6y = 24\):ewline\(8x - 6y = 24\)ewlineSubtract \(8x\) from both sides:ewline\(-6y = -8x + 24\)ewlineDivide by \(-6\):ewline\( y = \frac{4}{3} x - 4 \)ewlineFor the second equation \(4x - 3y = 12\):ewline\(4x - 3y = 12\)ewlineSubtract \(4x\) from both sides:ewline\(-3y = -4x + 12\)ewlineDivide by \(-3\):ewline\( y = \frac{4}{3} x - 4 \)
02

Plot the equations on the graph

Draw the graphs of each equation on the same coordinate plane. Both equations are \( y = \frac{4}{3}x - 4 \).
03

Identify the point of intersection

Examine the graphs. Since both lines are identical, they overlap completely, indicating an infinite number of solutions. The system is not merely inconsistent or having no solutions but has infinitely many solutions all along the line \( y = \frac{4}{3} x - 4 \).

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.

System of Equations
A system of equations consists of two or more equations with the same variables. The solution to a system of equations is the set of values for the variables that satisfies all the equations simultaneously.
To solve a system of equations, you can use various methods such as substitution, elimination, or the graphical method.
In this exercise, we focus on the graphical method, where each equation is represented as a line on a coordinate plane.
Slope-Intercept Form
The slope-intercept form of an equation is a way of writing linear equations. It is defined as:
\[ y = mx + b \]
In this formula, 'm' represents the slope of the line, and 'b' represents the y-intercept, where the line crosses the y-axis.
When given a system of equations, converting to slope-intercept form helps you plot the lines easily.
As seen in the exercise, the equations 8x - 6y = 24 and 4x - 3y = 12 were rewritten in slope-intercept form as:
\[ y = \frac{4}{3} x - 4 \]
This makes it straightforward to graph each line.
Infinite Solutions
A system of equations can have one solution, no solutions, or infinitely many solutions.
When dealing with the graphical method, you can determine the number of solutions by examining how the lines intersect:
1. If the lines intersect at one point, there is one solution.
2. If the lines are parallel and never intersect, there are no solutions.
3. If the lines are identical and overlap completely, there are infinitely many solutions.
In the exercise, both equations converted to the same line \( y = \frac{4}{3} x - 4 \), indicating that they overlap entirely, resulting in an infinite number of solutions along that line.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

The Lease-From-Us Company offers two different leasing plans for their top- ofthe-line color copying machine. The Economy plan costs \(\$ 175\) per month plus \(\$ 0.032\) per copy. The Standard plan costs \(\$ 225\) per month plus \(\$ 0.024\) per copy. (A) Write an equation for the monthly cost \(C\) of the Standard plan and the Economy for a month in which you make \(n\) copies. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(n\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine how many copies per month make it more economical to buy the Economy plan.

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. John goes into a donut shop and buys ten plain donuts and five cream-filled donuts for \(\$ 3.70 .\) Jane goes into the same shop and buys five plain donuts and ten cream-filled donuts for \(\$ 4.10 .\) What is the cost of a plain donut?

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A coffee wholesaler wishes to produce 60 pounds of a coffee blend selling at \(\$ 3.10\) per pound. How many pounds of coffee blends selling at \(\$ 3.35\) per pound and \(\$ 2.75\) per pound should be mixed to produce such a mixture?

Sketch a graph that represents the scenario described in the exercise. Be sure to clearly label any variables and the coordinate axes. Keep in mind that various graphs may be drawn to represent each situation. Suppose that the temperature of a metal bar starts at \(40^{\circ} \mathrm{C}\). Over the course of 30 minutes the temperature of the bar steadily increases to a temperature of \(160^{\circ} \mathrm{C} .\) It remains at that temperature for 45 minutes and then steadily cools off to a temperature of \(60^{\circ} \mathrm{C}\) over the next 20 minutes.

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} 2 r-5 s &=9 \\ s &=1-r \end{aligned}\right.$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.