A system of two linear equations with two variables is shown. $\newline$ $\begin{cases} 14 + x = 7y \ x - 7y = 14 \end{cases}$ $\newline$ Identify: $\newline$ A table for each equation. $\newline$ Equations in slope-intercept form. $\newline$ y - intercept and slope $\newline$ Graphs $\newline$ Solution(s)

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Grade 8

Write a linear equation from a slope and y-intercept

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Q. A system of two linear equations with two variables is shown.

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\begin{cases} 14 + x = 7y \ x - 7y = 14 \end{cases}

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Identify:

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A table for each equation.

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Equations in slope-intercept form.

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y - intercept and slope

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Graphs

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Solution(s)

Convert to Slope-Intercept Form: Convert the first equation to slope-intercept form. $\newline$ The first equation is $14 + x = 7y$ . To convert this to slope-intercept form $(y = mx + b)$ , we need to solve for $y$ . $\newline$ $14 + x = 7y$ $\newline$ Divide each term by $7$ to isolate $y$ . $\newline$ $y = (\frac{1}{7})x + 2$
Create Equation Table: Create a table for the first equation. $\newline$ We can choose arbitrary values for $x$ and calculate the corresponding $y$ values using the equation $y = \frac{1}{7}x + 2$ . $\newline$ Let's choose $x = -7, 0, 7$ . $\newline$ For $x = -7$ : $y = \frac{1}{7}(-7) + 2 = -1 + 2 = 1$ $\newline$ For $x = 0$ : $y = \frac{1}{7}(0) + 2 = 0 + 2 = 2$ $\newline$ For $x = 7$ : $y = \frac{1}{7}(7) + 2 = 1 + 2 = 3$
Convert Second Equation: Convert the second equation to slope-intercept form. $\newline$ The second equation is $x - 7y = 14$ . To convert this to slope-intercept form, we need to solve for $y$ . $\newline$ $x - 7y = 14$ $\newline$ Subtract $x$ from both sides. $\newline$ $-7y = -x + 14$ $\newline$ Divide each term by $-7$ to isolate $y$ . $\newline$ $y = \frac{1}{7}x - 2$
Create Second Table: Create a table for the second equation. $\newline$ We can use the same $x$ values as before to calculate the corresponding $y$ values using the equation $y = \frac{1}{7}x - 2$ . $\newline$ For $x = -7$ : $y = \frac{1}{7}(-7) - 2 = -1 - 2 = -3$ $\newline$ For $x = 0$ : $y = \frac{1}{7}(0) - 2 = 0 - 2 = -2$ $\newline$ For $x = 7$ : $y = \frac{1}{7}(7) - 2 = 1 - 2 = -1$
Identify Intercept and Slope: Identify the $y$ -intercept and slope for each equation. $\newline$ For the first equation, $y = \frac{1}{7}x + 2$ , the $y$ -intercept is $2$ and the slope is $\frac{1}{7}$ . $\newline$ For the second equation, $y = \frac{1}{7}x - 2$ , the $y$ -intercept is $-2$ and the slope is also $\frac{1}{7}$ .
Graph Equations: Graph the equations. $\newline$ We can graph the equations using the tables created in steps $2$ and $4$ . Each equation will be a straight line passing through the points listed in the tables.
Find Solution: Find the solution(s) to the system of equations. $\newline$ Since both equations have the same slope but different $y$ -intercepts, they are parallel lines and do not intersect. Therefore, there is no solution to this system of equations.