Question

problems 1-2: solve each system of equations.

1. $\

$$\begin{cases} y = 6x \\\\ 4x + y = 7 \\end{cases}$$

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1. $\

$$\begin{cases} y = 3x \\\\ x = -2y + 70 \\end{cases}$$

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problems 3-6: solve each system of equations. show or explain your thinking.

1. $\

$$\begin{cases} y = 3x - 2 \\\\ y = -2x + 8 \\end{cases}$$

$

1. $\

$$\begin{cases} y = -3x - 5 \\\\ y = 4x + 30 \\end{cases}$$

$

1. $\

$$\begin{cases} y = 2x - 9 \\\\ y = 4 + 2x \\end{cases}$$

$

1. $\

$$\begin{cases} x = 2 \\\\ y = 3x - 1 \\end{cases}$$

$

Explanation:

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Problem 1

Step1: Substitute $y=6x$ into second equation

$4x + 6x = 7$

Step2: Combine like terms and solve for $x$

$10x = 7 \implies x = \frac{7}{10} = 0.7$

Step3: Substitute $x$ to find $y$

$y = 6 \times 0.7 = 4.2$

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Problem 2

Step1: Substitute $y=3x$ into second equation

$x = -2(3x) + 70$

Step2: Simplify and solve for $x$

$x = -6x + 70 \implies 7x = 70 \implies x = 10$

Step3: Substitute $x$ to find $y$

$y = 3 \times 10 = 30$

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Problem 3

Step1: Set equations equal to each other

$3x - 2 = -2x + 8$

Step2: Solve for $x$

$3x + 2x = 8 + 2 \implies 5x = 10 \implies x = 2$

Step3: Substitute $x$ to find $y$

$y = 3(2) - 2 = 4$

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Problem 4

Step1: Set equations equal to each other

$-3x - 5 = 4x + 30$

Step2: Solve for $x$

$-3x - 4x = 30 + 5 \implies -7x = 35 \implies x = -5$

Step3: Substitute $x$ to find $y$

$y = -3(-5) - 5 = 10$

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Problem 5

Step1: Set equations equal to each other

$2x - 9 = 4 + 2x$

Step2: Simplify to check consistency

$2x - 2x = 4 + 9 \implies 0 = 13$

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Problem 6

Step1: Substitute $x=2$ into second equation

$y = 3(2) - 1$

Step2: Calculate $y$

$y = 6 - 1 = 5$

Answer:

1. $x=0.7$, $y=4.2$
2. $x=10$, $y=30$
3. $x=2$, $y=4$
4. $x=-5$, $y=10$
5. No solution (inconsistent system)
6. $x=2$, $y=5$