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Chapter 5: Problem 1

Verify whether the given ordered pair is a solution to the system. If it is not a solution, explain why not. a. \((-15.6,0.2)\) b. \((-4,23)\) c. \((2,12.3)\) \(\left\\{\begin{array}{l}y=47+3 x \\ y=8+0.5 x\end{array}\right.\) \(\left\\{\begin{array}{l}y=15-2 x \\ y=12+x\end{array}\right.\) \(\left\\{\begin{array}{l}y=4.5+5 x \\ y=2.3+5 x\end{array}\right.\) (?)

Short Answer

Expert verified
(Only pair a is a solution for system A.)

Step by step solution

01

Identify the Systems of Equations

There are three systems of equations provided: (1) System A: \(y = 47 + 3x\) and \(y = 8 + 0.5x\), (2) System B: \(y = 15 - 2x\) and \(y = 12 + x\), (3) System C: \(y = 4.5 + 5x\) and \(y = 2.3 + 5x\).
02

Check Solution a in System A

Substitute \((-15.6, 0.2)\) into System A. For the first equation, \(y = 47 + 3(-15.6)\), calculate the right side: \(47 - 46.8 = 0.2\). For the second equation, \(y = 8 + 0.5(-15.6)\), calculate the right side: \(8 - 7.8 = 0.2\). Both equations hold true, so \((-15.6, 0.2)\) is a solution for System A.
03

Check Solution a in System B

Substitute \((-15.6, 0.2)\) into System B. For the first equation, \(y = 15 - 2(-15.6)\), calculate the right side: \(15 + 31.2 = 46.2\). This does not equal \(0.2\), so \((-15.6, 0.2)\) is not a solution for System B.
04

Check Solution a in System C

Substitute \((-15.6, 0.2)\) into System C. For both equations, \(y = 4.5 + 5(-15.6)\) and \(y = 2.3 + 5(-15.6)\), calculate the right sides: \(-73.5\) and \(-75.7\), respectively. These are not equal to \(0.2\), so \((-15.6, 0.2)\) is not a solution for System C.
05

Check Solution b in System A

Substitute \((-4, 23)\) into System A. For the first equation, \(y = 47 + 3(-4)\), calculate the right side: \(47 - 12 = 35\). This does not equal \(23\), so \((-4, 23)\) is not a solution for System A.
06

Check Solution b in System B

Substitute \((-4, 23)\) into System B. For the first equation, \(y = 15 - 2(-4)\), calculate the right side: \(15 + 8 = 23\). For the second equation, \(y = 12 + (-4)\), calculate the right side: \(12 - 4 = 8\). The second equation does not match \(23\), so \((-4, 23)\) is not a solution for System B.
07

Check Solution b in System C

Substitute \((-4, 23)\) into System C. For both equations, \(y = 4.5 + 5(-4)\) and \(y = 2.3 + 5(-4)\), calculate the right sides: \(-15.5\) and \(-17.7\), respectively. These are not equal to \(23\), so \((-4, 23)\) is not a solution for System C.
08

Check Solution c in System A

Substitute \((2, 12.3)\) into System A. For the first equation, \(y = 47 + 3(2)\), calculate the right side: \(47 + 6 = 53\). This does not equal \(12.3\), so \((2, 12.3)\) is not a solution for System A.
09

Check Solution c in System B

Substitute \((2, 12.3)\) into System B. For the first equation, \(y = 15 - 2(2)\), calculate the right side: \(15 - 4 = 11\). This does not equal \(12.3\). Thus, \((2, 12.3)\) is not a solution for System B.
10

Check Solution c in System C

Substitute \((2, 12.3)\) into System C. For both equations, \(y = 4.5 + 5(2)\) and \(y = 2.3 + 5(2)\), calculate the right sides: \(14.5\) and \(12.3\), respectively. The second equation holds true, but the first does not, so \((2, 12.3)\) is not a solution for System C.

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Key Concepts

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

Algebraic Solutions
When solving systems of equations, an algebraic solution refers to the process of finding values for the variables that make both equations true at the same time. In the context of this exercise, we are provided with several systems, each comprising two equations. An algebraic solution is deemed valid for a particular ordered pair if, after substitution, both equations in the system equate to the same result for the given variables. The process involves calculating the expressions per each equation and ensuring they match the known values in the ordered pair.
This verifies the correctness of the algebraic solution, confirming whether the ordered pair satisfies the system.
Ordered Pairs
An ordered pair represents the potential solution to a system of equations, consisting generally of an \(x, y\) format. Each number in these pairs corresponds to the respective variable within the equations of the system. In the exercise, the ordered pairs \((-15.6, 0.2)\), \((-4, 23)\), and \((2, 12.3)\)\ represent potential solutions.
When checking whether these ordered pairs are solutions to their respective systems, we substitute them into the equations. We then check if the calculated values satisfy both equations simultaneously. If they do, the ordered pair is a solution. If not, it is not a solution and the ordered pair does not solve the system.
Understanding ordered pairs helps determine the relationship between variables in equations and finding where they intersect or hold true.
Substitution Method
The substitution method is a key strategy used to find solutions in systems of equations. This method involves substituting one variable, often solved for within one of the equations, into another equation. By doing so, we aim to simplify the system, making it easier to solve.
For example, in the given exercise, consider the system \[ y = 47 + 3x \] and \[ y = 8 + 0.5x \]. We can solve either equation for one variable, say \( y \), and substitute that expression into the other equation. This isolates one of the variables, allowing us to solve it algebraically. Once one variable is solved, we can use it to find the other by backing out from the initial substitution.
Substitution streamlines solving systems of equations, making it effective especially when the equations can easily be expressed in terms of one another.

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Most popular questions from this chapter

Solve each system of equations by elimination. Show your work. a. \(\left\\{\begin{array}{r}6 x+5 y=-20 \\ -6 x-10 y=25\end{array}\right.\) b. \(\left\\{\begin{array}{l}5 x-4 y=23 \\ 7 x+8 y=5\end{array}\right.\)

APPLICATION Hydroplanes are boats that move so fast they skim the top of the water. The hydroplane Spirit of the Tri-Cities qualified for the 2004 Columbia Cup race with a speed of \(145.000 \mathrm{mi} / \mathrm{h}\). The hydroplane Miss \(B\) qualified with a speed of \(163.162 \mathrm{mi} / \mathrm{h}\). (Northwest Hydro Racing, www.hydroracing.com) a. How long will each hydroplane take to run a 5 -lap race if one lap is \(2.5\) miles? b. Some boats limit the amount of fuel the motor burns to \(4.3\) gallons per minute. How much fuel will each boat use to run a 5 -lap race? c. Hydroplanes have a 50 -gallon tank though generally only about 43 gallons are put in. The rest of the tank is filled with foam to prevent sloshing. How many miles can each hydroplane go on one 43-gallon tank of fuel? d. Find each boat's fuel efficiency rate in miles per gallon.

Consider the system $$ \left\\{\begin{array}{l} y=7-3 x \\ y=11-2(x-5) \end{array}\right. $$ a. Convert each equation to the standard form \(a x+b y=c\). (a) b. Write a matrix for the system. (a)

APPLICATION The American College of Sports Medicine considers age as one factor when it recommends low and high heart rates during workout sessions. For safe and efficient training, your heart rate should be between \(55 \%\) and \(90 \%\) of the maximum heart rate level. The maximum heart rate is calculated by subtracting a person's age from 220 beats per minute. a. Define variables and write an equation relating age and maximum heart rate during workouts. b. Write a system of inequalities to represent the recommended high and low heart rates during a workout. (a) c. Graph the solution to show a region of safe and efficient heart rates for people of any age. d. What constraints should you add to limit your region to show the safe and efficient heart rates for people between the ages of 14 and 40 ? (a) e. Graph the new solution for \(8 \mathrm{~d}\).

In Example \(\mathrm{B}\), the inequality \(8-0.25 x<5\) was written to represent the situation where Erin slept less than 5 hours, and her sleep time was 8 hours minus \(0.25\) hour for each time the dog barked. However, Erin can't sleep less than 0 hours, so a more accurate statement would be the compound inequality \(0 \leq 8-0.25 x<5\). You can solve a compound inequality in the same way you've solved other inequalities; you just need to make sure you do the same operation to all three parts. Solve this inequality for \(x\) and graph the solution.
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