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Chapter 5: Problem 22

Solve each system by the addition method. \(\left\\{\begin{array}{l}3 x+2 y-14 \\ 3 x-2 y-10\end{array}\right.\)

Short Answer

Expert verified
The solutions to the system of equations are (1, 2), (1, -2), (-1,2), (-1, -2).

Step by step solution

01

Rearrange the equations

The first rearrangement can be to make both equations look more similar. Multiply the second equation by 2: \[4x^{2}-2y^{2}=-4\]. So, now the system to solve should look like: \[\left\{\begin{array}{l} 3x^{2}-2y^{2}=-5 \\ 4x^{2}-2y^{2}=-4 \end{array}\right..\]
02

Subtract second equation from the first

Subtract the second equation from the first will remove the \( y^2 \) term. This process looks like:\[\begin{align*}(3x^{2}-4x^{2}) - (2y^{2}-2y^{2}) & = -5 - (-4) \\-x^{2} & = -1\end{align*}\]This simplifies to \(x^{2}=1\).
03

Solve for x

Now, take the square roots on both sides of \(x^{2}=1\). The solutions would be \(x=1\) and \(x=-1\).
04

Substitute the values of x into original equations

Substitute \(x=1\) into the first original equation to solve for \(y\). The equation will be \(3(1)^2-2y^2=-5\), and simplifies to \(2y^2=3+5\), so \(y^2=4\). This gives \(y=2\) and \(y=-2\). Repeat this for \(x=-1\). So all together \(x=1, y=2\) and \(x=1, y=-2\) and \(x=-1, y=2\) and \(x=-1, y=-2\) are all solutions.

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Most popular questions from this chapter

Without graphing, in Exercises 73鈥�76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} (x-4)^{2}+(y+3)^{2} \leq 24 \\ (x-4)^{2}+(y+3)^{2} \geq 24 \end{array}\right.$$

Consider the objective function \(z-A x+B y \quad(A>0\) and \(B>0\) ) subject to the following constraints: \(2 x+3 y \leq 9, x-y \leq 2, x \geq 0,\) and \(y \geq 0 .\) Prove that the objective function will have the same maximum value at the vertices \((3,1)\) and \((0,3)\) if \(A-\frac{2}{3} B\).

This will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$f(x)=-2$$

Explain how to graph the solution set of a system of inequalities.

What is a system of linear inequalities?
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