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Chapter 5: Problem 56

Explain how to solve a nonlinear system using the substitution method. Use \(x^{2}+y^{2}=9\) and \(2 x-y=3\) to illustrate your explanation.

Short Answer

Expert verified
The solutions of the non-linear system are (2/5, -7/5) and (2, 1)

Step by step solution

01

Isolate one variable in one equation

From the given system of equations, take the linear equation \(2x - y = 3\) and isolate \(y\). When \(y\) is isolated, the equation becomes \(y = 2x - 3 \)
02

Substitute the isolated variable into the second equation

Substitute \(y\) from the first equation into the second equation \(x^2 + y^2 = 9\). This would give \[x^2 + (2x - 3)^2 = 9\].
03

Solve the equation

Expand, simplify, and factor out the resulting equation, \(x^2 + (2x - 3)^2 = 9\), to get \(5x^2 -12x+4=0\). Solve for \(x\), which gives two possible values, \(x=2/5\) or \(x=2\).
04

Substitute the values of \(x\) into equation 1 to get corresponding y-values

Substitute \(x = 2/5\) into \(y = 2x - 3 \) to obtain \(y = -7/5\). Alternatively, for \(x = 2\), the corresponding \(y\) becomes \(y = 1\). Therefore, the solutions of the given non-linear system of equations are \((2/5, -7/5)\) and \((2, 1)\)

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Most popular questions from this chapter

When is it easier to use the addition method rather than the substitution method to solve a system of equations?

In Exercises \(19-30,\) solve each system by the addition method. \(2 x+3 y=6\) \(2 x-3 y=6\)

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &2 x-3 y=-13\\\ &y=2 x+7 \end{aligned} $$ in- exercises-5-18-solve-each-system-by-the-substitution-method- beginaligned-2-x-3-y-13-y2-x7-endalig College Algebra Essentials 5 1 8 0 In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &2 x-3 y=8-2 x\\\ &3 x+4 y=x+3 y+14 \end{aligned} $$

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(9 x-3 y=12\) \(y=3 x-4\)

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) Cholesterol intake should be limited to \(300 \mathrm{mg}\) or less each day. One serving of scrambled eggs from McDonalds and one Double Beef Whopper from Burger King exceed this intake by 241 mg. Two servings of scrambled eggs and three Double Beef Whoppers provide \(1257 \mathrm{mg}\) of cholesterol. Determine the cholesterol content in each item.
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