/*! 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 51 Use a graphing utility to determ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 51

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{array}{l}y=-5 x+1 \\ y=x+3\end{array}\right.$$

Short Answer

Expert verified
When the lines of the equations are graphed, they intersect at one point, which means that the system of equations has one solution.

Step by step solution

01

Plot the first equation

First, plot the line of the equation \( y = -5x + 1 \). This is a linear equation, so it will be a straight line when graphed.
02

Plot the second equation

Next, plot the line of the equation \( y = x + 3 \). This is also a linear equation, and will be another straight line when graphed.
03

Identify the Intersection

Look at where the lines from Step 1 and Step 2 intersect. If they intersect at a single point, this is the solution to the system. If the lines do not intersect, then the system has no solutions. If the lines are the same line (they coincide), then there are infinite solutions.

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.

Linear Equations
Linear equations form the backbone of algebra. A linear equation is a type of equation where the highest power of the variable is one. In the context of our exercise, each equation represents a straight line when plotted on a graph. The general form for a linear equation in two variables is \( y = mx + b \), where:
  • \( m \) is the slope of the line, indicating its steepness or tilt.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
Both equations in the exercise, \( y = -5x + 1 \) and \( y = x + 3 \), fit this form. The differences in their slopes and intercepts determine how these lines behave relative to each other when graphed.
Graphing Utility
A graphing utility is a tool, often digital, that assists in plotting equations on a coordinate plane. These tools are incredibly helpful when dealing with systems of equations, as they provide a visual representation of the relationships between equations.

When you input the equations \( y = -5x + 1 \) and \( y = x + 3 \) into a graphing utility, it will graph each line based on their respective slopes and y-intercepts. The visualization helps in easily identifying where the lines meet, if at all.
  • Plot the equations accurately to see where they may intersect.
  • Different colors or styles may be used to distinguish between the lines.
Using a graphing utility can save time and reduce errors in plotting, especially with more complex equations.
Intersection
The intersection of two lines refers to the point where they cross each other on a graph. In terms of solving a system of linear equations, finding the intersection is crucial as it represents the solution to the system.

For our given equations, \( y = -5x + 1 \) and \( y = x + 3 \), the intersection point can be found by plotting both lines. The coordinates of this meeting point are the values of \( x \) and \( y \) that satisfy both equations simultaneously.
  • One intersection means there is a unique solution to the system.
  • No intersection implies the system has no solutions.
  • Countless intersections mean the lines are identical, indicating infinitely many solutions.
Solution of a System
The solution of a system of equations is the set of values that satisfy all equations involved. For linear equations, this solution is understood graphically as the intersection point(s) of the lines.

In our example with the system of equations \( y = -5x + 1 \) and \( y = x + 3 \), the solution is the point where these two lines intersect. By visualizing the system on a graphing utility, we identify the solution by seeing a single point where the lines cross.
  • If lines intersect at one point, the system has a singular solution.
  • If the lines run parallel without crossing, there鈥檚 no solution.
  • If lines overlap entirely, the solution includes all points on the line, indicating infinite solutions.
Understanding the solution helps in solving real-world problems where multiple conditions need to be met simultaneously.

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

Write a system of inequalities whose graphed solution set is an isosceles triangle.

Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \text { Model A } & \text { Model B } \\ \hline \text { Assembling } & 2 & 2.5 \\ \hline \text { Painting } & 4 & 1 \\ \hline \text { Packaging } & 1 & 0.75 \\ \hline \end{array} $$ The total times available for assembling, painting, and packaging are 4000 hours, 4800 hours, and 1500 hours, respectively. The profits per unit are \(\$ 50\) for model \(\mathrm{A}\) and \(\$ 75\) for model \(\mathrm{B}\). What is the optimal production level for each model? What is the optimal profit?

Investment You plan to invest up to $$\$ 30,000$$ in two different interest- bearing accounts. Each account is to contain at least $$\$ 6000$$. Moreover, one account should have at least twice the amount that is in the other account. (a) Find a system of inequalities that describes the amounts that you can invest in each account, and (b) sketch the graph of the system.

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 20 units of calcium, 10 units of iron, and 15 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 20 units of vitamin \(\mathrm{B}\). The minimum daily requirements for the diet are 400 units of calcium, 250 units of iron, and 220 units of vitamin B. (a) Find a system of inequalities describing the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that the dietitian can use in the diet. (b) Sketch the graph of the system. (c) A nutritionist normally gives a patient 18 ounces of food \(\mathrm{X}\) and \(3.5\) ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{X}\) are running low. What other combinations of foods \(\mathrm{X}\) and \(\mathrm{Y}\) can be given to the patient to meet the minimum daily requirements?

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=x+2 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 40 \\ x+y & \leq 30 \\ 2 x+3 y & \leq 65 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.