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Chapter 7: Problem 9

Solving a System by Substitution In Exercises \(7-14,\) solve the system by the method of substitution. Check your solution(s) graphically. $$\left\\{\begin{array}{c}{x-y=-4} \\ {x^{2}-y=-2}\end{array}\right.$$

Short Answer

Expert verified
The solutions to the system of equations are \(x = 2, y = -2\) and \(x = -5, y = -9\)

Step by step solution

01

Express one variable in terms of the other variable from the first equation

Express \(y\) in terms of \(x\) from the first equation. By simply adding \(y\) on both sides, the first equation becomes \(x = y + 4\)
02

Substitute \(y +4\) for \(x\) in the second equation

The second equation becomes \((y+4)^{2} - y = -2\). Expanding and simplifying, it yields \(y^{2} + 8y + 16 - y = -2\), which simplifies to \(y^{2} + 7y +18 = 0\)
03

Solve the equation for \(y\)

Given the simplified quadratic equation \(y^{2} + 7y +18 = 0\), we need to solve for \(y\). Factoring this quadratic equation, we obtain \((y + 2)(y + 9) = 0\). Setting each factor equal to zero gives \(y = -2\) and \(y = -9\) as the possible solutions for \(y\)
04

Find corresponding \(x\) values

Substituting \(y = -2\) into the equation \(x = y + 4\) yields \(x = 2\). Similarly, substituting \(y = -9\) into the same equation yields \(x = -5\). Therefore, the solutions to the system of equations are \(x = 2,\ y = -2\) and \(x = -5,\ y = -9\)

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique to solve systems of equations. It involves expressing one variable in terms of the other to simplify the system. For instance, given the system:
  • \(x - y = -4\)
  • \(x^2 - y = -2\)
First, solve the linear equation for one of the variables. Here, express \(x\) in terms of \(y\) from the first equation: \(x = y + 4\).
This expression can then be substituted into the other equation to reduce the number of variables. This makes it easier to solve, as you are left with just one variable in one equation.
Quadratic Equations
Quadratic equations typically take the form \(ax^2 + bx + c = 0\). They often appear when solving systems using substitution, like when substituting \(x = y + 4\) into \(x^2 - y = -2\) results in a quadratic equation:
\[y^2 + 7y + 18 = 0\]
This equation has powers as high as two, which categorizes it as a quadratic equation. Solving it requires techniques such as factoring or using the quadratic formula. Quadratic equations can show up in many applications, making them a crucial concept to master.
Factoring Polynomials
Factoring polynomials is a method used to solve quadratic equations by breaking them down into simpler expressions. For the equation \(y^2 + 7y + 18 = 0\), the polynomial can be factored to:
\((y + 2)(y + 9) = 0\)
This process involves finding two numbers that multiply to the constant term (18) and add up to the linear coefficient (7). In this case, those numbers are 2 and 9. Factoring zeroes out each component, giving solutions for \(y\):
  • \(y = -2\)
  • \(y = -9\)
Graphical Solution
A graphical solution involves plotting equations on a graph to visually find the intersection points. For our system:
  • The line \(x - y = -4\)
  • The parabola \(x^2 - y = -2\)

These plots will intersect at points representing the solutions of the system. In this scenario, the intersections at \((2, -2)\) and \((-5, -9)\) confirm our algebraic solutions. Using graphing, you can visually verify solutions and understand the relationship between equations.

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Most popular questions from this chapter

Investment Portfolio A total of \(\$ 24,000\) is invested in two corporate bonds that pay 3.5\(\%\) and 5\(\%\) simple interest. The investor wants an annual interest income of \(\$ 930\) from the investments. What amount should be invested in the 3.5\(\%\) bond?

Optimal Revenue An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges \(\$ 1600\) for an audit and \(\$ 250\) for a tax return. Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue?

Writing Describe two ways of solving for the constants in a partial fraction decomposition.

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Advanced Applications In Exercises 69 and 70 , solve the system of equations for \(u\) and \(v .\) While solving for these variables, consider the transcendental functions as constants. (Systems of this type appear in a course in differential equations.) $$ \left\\{\begin{aligned} u \cos 2 x+& v \sin 2 x=& 0 \\ u(-2 \sin 2 x)+v(2 \cos 2 x) &=\csc 2 x \end{aligned}\right. $$
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