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Chapter 7: Problem 32

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}3 x-7 y=13 \\ 6 x+5 y=7\end{array}\right.\)

Short Answer

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

Step by step solution

01

Multiply The Equations To Make Coefficients Opposite

Choose one of the variables to eliminate. In this case, the variable 'y' is chosen. Subtracting the two equations will eliminate 'y'. However, before this can be done, the equations need to be multiplied to make the coefficients of 'y' opposite in each equation. Multiply the first equation by 5 and the second equation by 7. The equations become: \(15x - 35y = 65\) \(42x + 35y = 49\)
02

Implement The Addition Method

Now, add the two equations together. This will eliminate 'y'.\(15x + 42x = 65 + 49\) Simplify to get: \(57x = 114\)
03

Solve For x

Solving for 'x' involves dividing each side of the equation by 57.\(x = 114/57\)Simplify to get: \(x = 2\)
04

Substitute x Into The Original Equation

Substitute \(x = 2\) into one of the original equations to solve for 'y'. Choosing the first equation gives:\(3x - 7y = 13\)Substitute x to get:\(3*2 - 7y = 13\)which simplifies to: \(6 - 7y = 13\)
05

Solve For y

Solving for 'y' involves first subtracting 6 from each side of the equation, and then dividing each side by -7. So, \(-7y = 13 - 6\) or \(-7y = 7\) Dividing each side by -7 gives \(y = -1\)
06

Check The Proposed Solution

Substitute \(x = 2\) and \( y = -1\) into the original system of equations and ensure that both equations are true. \( (3*2) - (7*-1) = 13\) and \( (6*2) + (5*-1) = 7\)Both equations are true, so \(x = 2\) and \( y = -1\) is the solution to the system.

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Most popular questions from this chapter

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$ f(x)=62+35 \log (x-4), $$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve Exercises 37-38. a. According to the model, what percentage of her adult height has a girl attained at age ten? Use a calculator with a LOG key and round to the nearest tenth of a percent. b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15 , inclusive?

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}2 x+y<3 \\ x-y>2\end{array}\right.\)

Describe the shape of a scatter plot that suggests modeling the data with a quadratic function.

Graph each linear inequality. \(x-y \leq 1\)

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed the solution set of \(y \geq x+2\) and \(x \geq 1\) without using test points.
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