/*! 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 7 Determine if the ordered pair is... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 7

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(3 x-5 y=-7\) \(x-4 y=-7\) a. (1,2) b. \(\left(-\frac{2}{3}, 1\right)\)

Short Answer

Expert verified
(1,2) is a solution. (-2/3,1) is not a solution.

Step by step solution

01

Substitute the values of (1,2) into the first equation

Substitute the values of x and y from the ordered pair (1,2) into the first equation. dFirst equation: \(3x - 5y = -7\). Substituting, \(3(1) - 5(2) = -7\). Calculate to check if the equation holds true: \(3 - 10 = -7\). Since \(-7 = -7\), the first equation is satisfied.
02

Substitute the values of (1,2) into the second equation

Substitute the values of x and y from the ordered pair (1,2) into the second equation. Second equation: \(x - 4y = -7\). Substituting, \(1 - 4(2) = -7\). Calculate to check if the equation holds true: \(1 - 8 = -7\). Since \(-7 = -7\),the second equation is satisfied.
03

Verify if (1,2) is a solution

Since (1,2) satisfies both equations, (\(3x - 5y = -7\) and \(x - 4y = -7\)), (1,2) is a solution to the system.
04

Substitute the values of (-2/3, 1) into the first equation

Substitute the values of x and y from the ordered pair (-2/3, 1) into the first equation. First equation: \(3x - 5y = -7\). Substituting, \(3(-\frac{2}{3}) - 5(1) = -7\). Calculate to check if the equation holds true: \(-2 - 5 = -7\). Since \(-7 = -7\), the first equation is satisfied.
05

Substitute the values of (-2/3, 1) into the second equation

Substitute the values of x and y from the ordered pair (-2/3, 1) into the second equation. Second equation: \(x - 4y = -7\). Substituting, \(-\frac{2}{3} - 4(1) = -7\). Calculate to check if the equation holds true: \(-\frac{2}{3} - 4 = -\frac{2}{3} - 4 = -7\) . Since \(-\frac{2}{3} - 4\) is not equal to \(-7\), the second equation is not satisfied.
06

Verify if (-2/3, 1) is a solution

Since (-2/3, 1) does not satisfy both equations, (\(3x - 5y = -7\) and \(x - 4y = -7\)), (-2/3, 1) is not a solution to the system.

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 are algebraic expressions where each term is a constant or the product of a constant and a single variable. For example, in the equations provided like 3x - 5y = -7 and x - 4y = -7, each term is either a constant or involves a variable (x or y) raised to the first power.
Linear equations are fundamental in algebra and can be used to represent various real-world scenarios, such as predicting costs or modeling relationships between variables.
To solve a system of linear equations means finding a set of values for the variables that make all the equations true at the same time.
Substitution Method
The substitution method is a technique used to solve systems of equations. In this method, you solve one of the equations for one variable and then substitute that expression into the other equation. This helps in finding the values of the variables sequentially.
Steps for solving using the substitution method are as follows:
  • Solve one of the equations for one of the variables.
  • Substitute this expression into the other equation.
  • Solve the resulting equation for the remaining variable.
  • Finally, substitute back to find the value of the first variable.
In the example, we substituted the values directly from the ordered pairs into the equations to check their validity, rather than isolating variables and substituting algebraically.
Ordered Pairs
Ordered pairs are a way to represent solutions to equations with two variables, typically written as (x, y). They show the relationship between the values of x and y that satisfy the equation.
In the given exercise, you were asked to determine if the ordered pairs (1, 2) and (-2/3, 1) are solutions to the system of equations provided.
Checking if an ordered pair is a solution involves substituting the x and y values into each equation:
  • If both substitutions satisfy their respective equations, the ordered pair is a solution to the system.
  • If even one substitution does not satisfy its equation, the ordered pair is not a solution.
Working through these steps systematically helps ensure you find accurate results and verify solutions correctly.

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

Systems of equations play an important role in the analysis of supply and demand. For example, suppose that a theater company wants to set an optimal price for tickets to a show. If the theater sells tickets for \(\$ 1\) each, there would be a high demand and many people would buy tickets. However, the revenue brought in would not cover the expense of the show. Therefore, the theater is not willing to offer (supply) tickets at this low price. If the theater sells tickets for \(\$ 10,000\) each, chances are that no one would buy a ticket (demand would be low). In an open market, the price of an item is dependent on the demand by consumers and the supply offered by producers. Competition between buyers and sellers steers the price toward an equilibrium price-that is, the price where supply equals demand. The number of items offered and sold at the equilibrium price is the equilibrium quantity. Use this information for Exercises \(87-88 .\) The price \(p\) (in \(\$$ ) of a cookbook is determined by the number of cookbooks \)x\( demanded by consumers and supplied by the publisher. Supply: \)\quad p=0.002 x\( Demand: \)\quad p=-0.005 x+70$ a. Solve the system of equations defined by the supply and demand models. b. What is the equilibrium price? c. What is the equilibrium quantity?

Write an inequality to represent the given statement. The value of \(y\) does not exceed three times the value of \(x\).

Determine whether the ordered pair is a solution to the inequality. \(2 x+3 y>6\) a. (-3,3) b. (5,-1) c. (0,2)

Josh makes \(\$ 24 /\) hr tutoring chemistry and \(\$ 20 / \mathrm{hr}\) tutoring math. Let \(x\) represent the number of hours per week he spends tutoring chemistry. Let \(y\) represent the number of hours per week he spends tutoring math. a. Write an objective function representing his weekly income for tutoring \(x\) hours of chemistry and \(y\) hours of math. b. The time that Josh devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints. \- The number of hours spent tutoring each subject cannot be negative. \- Due to the academic demands of his own classes he tutors at most \(18 \mathrm{hr}\) per week. \- The tutoring center requires that he tutors math at least 4 hr per week. \- The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many hours tutoring math and how many hours tutoring chemistry should Josh work to maximize his income? g. What is the maximum income? h. Explain why Josh's maximum income is found at a point on the line \(x+y=18\).

a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of \(x\) and \(y\) that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. $$ \begin{array}{l} x \geq 0, y \geq 0 \\ x+y \leq 60 \\ y \leq 2 x \\ \text { Maximize: } z=250 x+150 y \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision 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.