Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 5: Problem 2

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{aligned} 2 x-y &=2 \\ x+3 y &=8 \end{aligned}\right.\) (a) \((2,1)\) (b) \((2,2)\)

Short Answer

Expert verified
The ordered pair (2,1) is not a solution of the system. The ordered pair (2,2) is a solution of the system.

Step by step solution

01

Given System of Equations

The given system of equations to check is \[ \left\{\begin{aligned} 2 x-y &=2 \ x+3 y &=8 \end{aligned}\right. \]
02

Check for the ordered pair (2,1)

For the ordered pair (2,1), substitute x with 2 and y with 1 in both equations and check if the equations hold true.\[ 2x - y = 2*2 - 1 = 3 \] and not equal to 2. This shows that the ordered pair (2,1) is not a solution to the system.
03

Check for the ordered pair (2,2)

For the ordered pair (2,2), substitute x with 2 and y with 2 in both equations and check if equations hold true. \[2x - y = 2*2 - 2 = 2,\] which is equal to 2. For the second equation, \[x + 3y = 2 + 3*2 = 8,\] which is also equal to 8. Therefore, the ordered pair (2,2) is 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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Ordered Pair Solution
Understanding the concept of an ordered pair solution is crucial when solving systems of equations. An ordered pair, typically written as (x, y), represents potential values for the variables x and y that may satisfy both equations simultaneously. To verify whether an ordered pair is a solution, we substitute the x and y values into each equation. If both equations are satisfied—meaning they produce true statements—the ordered pair is indeed a solution.

From the exercise given, to determine if (2,1) is a solution, we replaced x with 2 and y with 1 in the equations. However, this did not satisfy the first equation, hence (2,1) is not a solution. Conversely, the ordered pair (2,2) satisfied both equations, thus it is a legitimate solution. Remember that for a pair to qualify as a solution, both equations must hold true after substitution. If either equation does not match, the pair isn't a solution.
Substitution Method
The substitution method is a widely used technique for solving a system of equations algebraically. In this method, we express one variable in terms of the other using one of the equations, and then substitute this expression into the other equation. This substitution allows us to solve for one variable. Once we've found the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable.

While the original exercise does not require using the substitution method to find the ordered pair solution, it's a good skill to master. To start, you could isolate y from the equation 2x - y = 2 to get y = 2x - 2 and then substitute this into the second equation, x + 3y = 8. This way, you can find the values for x and y that satisfy both equations without guessing or checking potential solutions.
Algebraic Solution
When we talk about an algebraic solution to a system of equations, we refer to solving the system by manipulation of the equations using algebraic methods, such as substitution, elimination, or graphing. The objective is to find the values of the variables that satisfy all equations in the system. These values make up the solution set.

For the system in question, the algebraic solution involves checking given ordered pairs instead of solving for them. However, if we were to solve the system, we would use algebraic methods such as the substitution method mentioned earlier. By applying these techniques, we would arrive at a precise solution for x and y, just as we confirmed the validity of the pair (2,2) using the values given. Algebraic solutions provide us with a systematic and logical approach to determining the exact values that solve the system of equations.

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

Optimal Revenue An accounting firm charges $$\$ 2500$$ for an audit and $$\$ 350$$ for a tax return. Research and available resources have indicated the following constraints. \- The firm has 900 hours of staff time available each week. \- The firm has 155 hours of review time available each week. \- Each audit requires 75 hours of staff time and 10 hours of review time. \- Each tax return requires \(12.5\) hours of staff time and \(2.5\) hours of review time. What numbers of audits and tax returns will bring in an optimal revenue?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \leq \sqrt{3 x}+1 \\ y \geq x+1\end{array}\right.$$

MAKE A DECISION: STOPPING DISTANCE In testing of the new braking system of an automobile, the speed (in miles per hour) and the stopping distance (in feet) were recorded in the table below. $$ \begin{array}{|c|c|} \hline \text { Speed, } x & \text { Stopping distance, } y \\ \hline 30 & 54 \\ \hline 40 & 116 \\ \hline 50 & 203 \\ \hline 60 & 315 \\ \hline 70 & 452 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}5 c+250 b+13,500 a=1140 \\ 250 c+13,500 b+775,000 a=66,950 \\ 13,500 c+775,000 b+46,590,000 a=4,090,500\end{array}\right.\) (b) Use the regression feature of a graphing utility to check your answer to part (a). (c) A car design specification requires the car to stop within 520 feet when traveling 75 miles per hour. Does the new braking system meet this specification?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{rr}x^{2}+y \leq & 6 \\ x \geq & -1 \\ y \geq & 0\end{array}\right.$$

Sailboats The total numbers \(y\) (in thousands) of sailboats purchased in the United States in the years 2001 to 2005 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to \(2003 .\) (Source: National Marine Manufacturers Association) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \text { Number, } y \\ \hline-2 & 18.6 \\ \hline-1 & 15.8 \\ \hline 0 & 15.0 \\ \hline 1 & 14.3 \\ \hline 2 & 14.4 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{aligned} 5 c &+10 a=78.1 \\\ 10 b &=-9.9 \\ 10 c &+34 a=162.1 \end{aligned}\right.\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. Compare the quadratic model with the model found in part (a).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.

Sign-up for free