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Chapter 7: Problem 24

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{aligned}-4 x+y &=-11 \\ 2 x-3 y &=5 \end{aligned}\right.\)

Short Answer

Expert verified
The solution to the system of equations is \(x = -2\) and \(y = -19\).

Step by step solution

01

Solve one of the equations for one variable

From equation 1, if we solve for y, we get \(y = -4x - 11\).
02

Substitute y into the second equation

Substitute \(y = -4x - 11\) into equation 2: \(2x -3(-4x -11) = 5\). This simplifies to \(2x + 12x + 33 = 5\), or \(14x + 33 = 5\).
03

Solve for x

Subtract 33 from both sides, resulting in \(14x = -28\). Divide both sides by 14, finding \(x = -2\).
04

Substitute x into the first equation

Substitute \(x = -2\) into equation 1: \(-4(-2) + y = -11\). This simplifies to \(8 + y = -11\).
05

Solve for y

Subtract 8 from both sides, results in \(y = -19\).
06

Checking the solution

Substitute \(x = -2\) and \(y = -19\) into both original equations to confirm the solution. For equation 1: \(-4(-2) + (-19) = -11\) and for equation 2: \(2(-2) - 3(-19) = 5\). Both equalities hold, thus the solution is valid.

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Key Concepts

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

Systems of Equations
A system of equations consists of two or more equations with the same set of variables. In our problem, it involves two linear equations, each containing both variables, \(x\) and \(y\). Solving a system means finding a set of values for the variables that make all the equations true simultaneously. These equations represent lines on a graph, and the solution is where the lines intersect. Systems can have:
  • No solution (parallel lines that never meet)
  • One solution (intersecting lines at one point)
  • Infinite solutions (the same line, or coincidental lines)
In this problem, we look to find that one point of intersection where both equations are satisfied.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. In the substitution method, you isolate one variable in one equation, as we did by solving for \(y\) in the first equation to get \(y = -4x - 11\). Then, you substitute this expression into the other equation.

For instance, substituting \(y = -4x - 11\) into the second equation allows us to eliminate \(y\) and solve for \(x\). This step-by-step simplification involves combining like terms and following arithmetic operations, which is key for mastering algebraic manipulation skills.
Solution Verification
Solution verification is crucial to ensure the accuracy of your solution. Once you find a potential set of values for \(x\) and \(y\), substitute them back into the original equations to ensure that they satisfy both.

In our case, both \(x = -2\) and \(y = -19\) need to satisfy the original equations:
  • For equation 1: \(-4(-2) + (-19)\) should equal \(-11\)
  • For equation 2: \(2(-2) - 3(-19)\) should equal \(5\)
Double-checking like this confirms that the solution is not just correctly computed but also relevant and applicable to the problem at hand.
Linear Equations
Linear equations are algebraic expressions where the highest power of the variable is one. The general form is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

These equations predict straight-line graphs, hence the name 'linear'. Exploring linear equations allows understanding of core algebraic concepts and graphical interpretations. Recognizing their structure simplifies the process of isolating variables and performing substitutions, such as when we solved for \(y\) in terms of \(x\) in the first equation. Linear equations form the basic foundation for more complex algebra and calculus concepts.

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Most popular questions from this chapter

The value of \(a\) in \(y=a x^{2}+b x+c\) and the vertex of the parabola are given. How many \(x\)-intercepts does the parabola have? Explain how you arrived at this number. \(a=1 ;\) vertex at \((2,0)\)

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(f(x)=x^{2}-2 x-8\)

a. A student earns \(\$ 15\) per hour for tutoring and \(\$ 10\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that describes total weekly earnings. b. The student is bound by the following constraints: \- To have enough time for studies, the student can work no more than 20 hours per week. \- The tutoring center requires that each tutor spend at least three hours per week tutoring. \- The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that describes these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17)\), and \((8,12)\).] e. Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for hours per week and working as a teacher's aide for hours per week. The maximum amount that the student can earn each week is $\$$

Members of the group should interview a business executive who is in charge of deciding the product mix for a business. How are production policy decisions made? Are other methods used in conjunction with linear programming? What are these methods? What sort of academic background, particularly in mathematics, does this executive have? Present a group report addressing these questions, emphasizing the role of linear programming for the business.

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x+y \leq 4 \\ y \geq 2 x-4\end{array}\right.\)
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