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Chapter 7: Problem 5

Determine whether each ordered pair is a solution of the system. $$\left\\{\begin{array}{l} 2 x-y=4 \\ 8 x+y=-9 \end{array}\right.$$ (a) (0,-4) (b) (-2,7) (c) \(\left(\frac{3}{2},-1\right)\) (d) \(\left(-\frac{1}{2},-5\right)\)

Short Answer

Expert verified
Only the ordered pairs (-2,7) and \(\left(-\frac{1}{2},-5\right)\) are solutions to the given system of equations.

Step by step solution

01

Check the first ordered pair (0,-4)

To check whether the ordered pair (0,-4) is a solution of the system, replace x with 0 and y with -4 in the system of equations. If both equations are valid, the ordered pair is a solution, otherwise it's not. In the first equation: \(2(0) - (-4) = 4\) holds true, and in the second one: \(8(0) + (-4) = -4\) doesn't hold true. Thus, ordered pair (0,-4) is not a solution of the system.
02

Check the second ordered pair (-2,7)

Replace x with -2 and y with 7 in the system. Result for the first equation would be: \(2(-2) - 7 = -11\) and for the second: \(8(-2) + 7 =-9\). The ordered pair (-2,7) satisfies both equations, hence it is a solution of the system.
03

Check the third ordered pair \(\left(\frac{3}{2},-1\right)\)

Substitute x with \(\frac{3}{2}\) and y with -1 in the system. We find that \(2(\frac{3}{2}) - (-1)\) is not equal to 4, and \(8(\frac{3}{2}) + (-1)\) is not equal to -9. So, \(\left(\frac{3}{2},-1\right)\) is not a solution of the system.
04

Check the fourth ordered pair \(\left(-\frac{1}{2},-5\right)\)

Put x as \(-\frac{1}{2}\) and y as -5 in the system. Evaluating gives us \(2(-\frac{1}{2}) - (-5)\) equates to 4 and \(8(-\frac{1}{2}) + (-5)\) equates to -9. Thus, \(\left(-\frac{1}{2},-5\right)\) is a solution of the system.

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Key Concepts

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

Ordered Pair
An ordered pair is a fundamental concept in mathematics, especially in coordinate systems, where it represents a pair of numbers indicating a specific location. Mathematically, an ordered pair is written as \( (x, y) \) where \( x \) and \( y \) are numbers that correspond to the horizontal (x-axis) and vertical (y-axis) positions, respectively.

When dealing with systems of equations, an ordered pair is used as a potential solution. To verify whether the ordered pair truly satisfies the system, each number of the pair is substituted into the given equations. If the equations are true for both values, then the ordered pair is indeed a solution to the system, representing a point where the graphed lines would intersect on a coordinate plane.
Substitution Method
The substitution method is a technique used to find algebraic solutions for systems of equations. It involves substituting one variable with an equivalent expression from another equation which allows the equations to be solved for one variable at a time.

Applying the Substitution Method

  • First, solve one of the equations for one variable in terms of the other.
  • Substitute this expression into the other equation.
  • Solve for the remaining variable.
  • Use the value of this variable to find the value of the other.

The substitution method is especially valuable when you can easily express one variable in terms of the other or when the equations are conducive to straightforward manipulation.
Algebraic Solutions
An algebraic solution refers to the precise answers obtained when solving equations or systems of equations. Such solutions are found using algebraic methods including substitution, elimination, graphing, or matrix operations. In the context of systems of equations, the algebraic solution usually comes in the form of ordered pairs, indicating the specific values of variables that satisfy all equations simultaneously.

Achieving an algebraic solution typically includes the following steps:
  • Setting up the proper equations based on the problem.
  • Manipulating the equations using algebraic techniques.
  • Solving the equations to find the values of the unknown variables.
  • Checking the solutions back into the original equations to validate them.
Precalculus
Precalculus is a branch of mathematics that prepares students for the study of calculus, focusing on concepts and skills that are foundational for understanding calculus topics. It includes a wide range of topics such as algebra, geometry, trigonometry, and analytical geometry.

Importance in Systems of Equations

Systems of equations are a crucial topic in precalculus. They embody the understanding of how multiple equations can relate to one another and the concept of finding common solutions that satisfy all equations within the system. Precalculus helps students develop the problem-solving skills necessary to tackle complex systems of equations, reassuring that the foundational knowledge is in place for more advanced study.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. For the rational expression \(\frac{2 x+3}{x^{2}(x+2)^{2}},\) the partial fraction decomposition is of the form \(\frac{A x+B}{x^{2}}+\frac{C x+D}{(x+2)^{2}}\)

In testing a new automobile braking system, engineers recorded the speed \(x\) (in miles per hour) and the stopping distance \(y\) (in feet) in the following table. $$\begin{array}{|l|c|c|c|} \hline \text { Speed, } x & 30 & 40 & 50 \\\ \hline \text { Stopping Distance, } y & 55 & 105 & 188 \\ \hline \end{array}$$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the system. $$\left\\{\begin{array}{rr} 3 c+120 b+5000 a= 348 \\\ 120 c+5000 b+216,000 a= 15,250 \\ 5000 c+216,000 b+9,620,000 a=687,500 \end{array}\right.$$ (b) Graph the parabola and the data on the same set of axes. (c) Use the model to estimate the stopping distance when the speed is 70 miles per hour.

Use a graphing utility to graph the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. Objective function: \(z=3 x+y\) Constraints: $$\begin{aligned}x & \geq 0 \\\y & \geq 0 \\ x+4 y & \leq 60 \\\3 x+2 y & \geq 48\end{aligned}$$

Identify the graph of the rational function and the graph representing each partial fraction of its partial fraction decomposition. Then state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs representing the partial fractions of the decomposition. To print an enlarged copy of the graph, go to MathGraphs.com. $$\begin{aligned} &\text { (a) } y=\frac{x-12}{x(x-4)}\\\ &=\frac{3}{x}-\frac{2}{x-4} \end{aligned}$$ $$\begin{aligned} &\text { (b) } y=\frac{2(4 x-3)}{x^{2}-9}\\\ &=\frac{3}{x-3}+\frac{5}{x+3} \end{aligned}$$

Find the equation of the parabola $$y=a x^{2}+b x+c$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $$(1,3),(2,2),(3,-3)$$
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