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Chapter 7: Problem 96

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

Short Answer

Expert verified
The solution to the system of equations is \(x = -1\) and \(y = 2/3\).

Step by step solution

01

Multiply the equations

To facilitate the task of eliminating one of the variables by addition, the equations can be multiplied by suitable numbers. For this particular problem, to eliminate variable y, equation 1 can be multiplied by 3 and equation 2 by 5. Thus, the equations become: \n\[15x + 9x = 0,\n15x - 10y = -6.\]
02

Add the equations

After the equations have been multiplied, the aim is to add them together in such a way, that one of the variables cancels out. If we subtract equation 2 from equation 1, we eliminate y and obtain: \n\[-6x = 6,\]
03

Solve for first variable

After simplifying, you are left with a much simpler, one-variable equation. Dividing both sides by -6, you obtain: \n\[x = -1.\]
04

Substitute x into original equation

Once you have an answer for x, you can substitute that value back into one of the original equations (in this case, equation 2 simplifies more readily) to find y: \n\[2*(-1) + 3y = 0,\]
05

Solve for second variable

With the value for x substituted in, you again have a single-variable equation, and you can solve for y by just isolating it on one side, and get \n\[y = 2/3.\]

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