Question Find the $y$ -value of the solution to the system of equations created by the line $3x-2y=-4$ and the line passing through the points $(-3,-9), (-1,-5), (3,3), (5,7)$ .

Studdy Solution

STEP 1

Assumptions
1. The first equation of the system is $3x - 2y = -4$ .
2. The second equation of the system can be derived from the data in the table.
3. The data in the table represents points that lie on the line of the second equation.
4. To find the $y$ -value of the solution to the system, we need to solve the system of equations.

STEP 2

We will use the two points from the table to find the slope of the second line. Let's use the points $(-3, -9)$ and $(3, 3)$ .
$slope = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 3

Now, plug in the values for $x_1, y_1, x_2, y_2$ to calculate the slope.
$slope = \frac{3 - (-9)}{3 - (-3)}$

STEP 4

Simplify the numerator and denominator.
$slope = \frac{3 + 9}{3 + 3}$
$slope = \frac{12}{6}$

STEP 5

Calculate the slope.
$slope = 2$

STEP 6

Now that we have the slope, we can use the point-slope form to write the equation of the second line. We can use any point from the table; let's use the point $(-3, -9)$ .
$y - y_1 = slope \cdot (x - x_1)$

STEP 7

Plug in the values for $x_1, y_1$ and the slope into the point-slope form.
$y - (-9) = 2 \cdot (x - (-3))$

STEP 8

Simplify the equation.
$y + 9 = 2 \cdot (x + 3)$

STEP 9

Distribute the slope on the right side of the equation.
$y + 9 = 2x + 6$

STEP 10

Subtract 9 from both sides to get the equation in slope-intercept form, $y = mx + b$ .
$y = 2x + 6 - 9$

STEP 11

Simplify the equation.
$y = 2x - 3$

STEP 12

Now we have the second equation of the system. The system of equations is:
$\begin{cases} 3x - 2y = -4 \\ y = 2x - 3 \end{cases}$

STEP 13

To find the solution to the system, we can substitute the second equation into the first equation.
$3x - 2(2x - 3) = -4$

STEP 14

Distribute the $-2$ on the left side of the equation.
$3x - 4x + 6 = -4$

STEP 15

Combine like terms.
$-x + 6 = -4$

STEP 16

Add $x$ to both sides to isolate the $x$ term on one side.
$6 = x - 4$

STEP 17

Add 4 to both sides to solve for $x$ .
$x = 6 + 4$

STEP 18

Calculate the value of $x$ .
$x = 10$

STEP 19

Now that we have the value of $x$ , we can substitute it back into the second equation to find the value of $y$ .
$y = 2(10) - 3$

STEP 20

Calculate the value of $y$ .
$y = 20 - 3$

STEP 21

Simplify to get the final value of $y$ .
$y = 17$
The $y$ -value of the solution to the system is 17.

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Question Find the yyy-value of the solution to the system of equations created by the line 3x−2y=−43x-2y=-43x−2y=−4 and the line passing through the points (−3,−9),(−1,−5),(3,3),(5,7)(-3,-9), (-1,-5), (3,3), (5,7)(−3,−9),(−1,−5),(3,3),(5,7).

Studdy Solution

STEP 1

STEP 2

We will use the two points from the table to find the slope of the second line. Let's use the points (−3,−9)(-3, -9)(−3,−9) and (3,3)(3, 3)(3,3).slope=y2−y1x2−x1slope = \frac{y_2 - y_1}{x_2 - x_1}slope=x2​−x1​y2​−y1​​

STEP 3

Now, plug in the values for x1,y1,x2,y2x_1, y_1, x_2, y_2x1​,y1​,x2​,y2​ to calculate the slope.slope=3−(−9)3−(−3)slope = \frac{3 - (-9)}{3 - (-3)}slope=3−(−3)3−(−9)​

STEP 4

Simplify the numerator and denominator.slope=3+93+3slope = \frac{3 + 9}{3 + 3}slope=3+33+9​slope=126slope = \frac{12}{6}slope=612​

STEP 5

Calculate the slope.slope=2slope = 2slope=2

STEP 6

Now that we have the slope, we can use the point-slope form to write the equation of the second line. We can use any point from the table; let's use the point (−3,−9)(-3, -9)(−3,−9).y−y1=slope⋅(x−x1)y - y_1 = slope \cdot (x - x_1)y−y1​=slope⋅(x−x1​)

STEP 7

Plug in the values for x1,y1x_1, y_1x1​,y1​ and the slope into the point-slope form.y−(−9)=2⋅(x−(−3))y - (-9) = 2 \cdot (x - (-3))y−(−9)=2⋅(x−(−3))

STEP 8

Simplify the equation.y+9=2⋅(x+3)y + 9 = 2 \cdot (x + 3)y+9=2⋅(x+3)

STEP 9

Distribute the slope on the right side of the equation.y+9=2x+6y + 9 = 2x + 6y+9=2x+6

STEP 10

Subtract 9 from both sides to get the equation in slope-intercept form, y=mx+by = mx + by=mx+b.y=2x+6−9y = 2x + 6 - 9y=2x+6−9

STEP 11

Simplify the equation.y=2x−3y = 2x - 3y=2x−3

STEP 12

Now we have the second equation of the system. The system of equations is:{3x−2y=−4y=2x−3\begin{cases} 3x - 2y = -4 \\ y = 2x - 3 \end{cases}{3x−2y=−4y=2x−3​

STEP 13

To find the solution to the system, we can substitute the second equation into the first equation.3x−2(2x−3)=−43x - 2(2x - 3) = -43x−2(2x−3)=−4

STEP 14

Distribute the −2-2−2 on the left side of the equation.3x−4x+6=−43x - 4x + 6 = -43x−4x+6=−4

STEP 15

Combine like terms.−x+6=−4-x + 6 = -4−x+6=−4

STEP 16

Add xxx to both sides to isolate the xxx term on one side.6=x−46 = x - 46=x−4

STEP 17

Add 4 to both sides to solve for xxx.x=6+4x = 6 + 4x=6+4

STEP 18

Calculate the value of xxx.x=10x = 10x=10

STEP 19

Now that we have the value of xxx, we can substitute it back into the second equation to find the value of yyy.y=2(10)−3y = 2(10) - 3y=2(10)−3

STEP 20

Calculate the value of yyy.y=20−3y = 20 - 3y=20−3

STEP 21

Simplify to get the final value of yyy.y=17y = 17y=17The yyy-value of the solution to the system is 17.

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Question Find the $y$ -value of the solution to the system of equations created by the line $3x-2y=-4$ and the line passing through the points $(-3,-9), (-1,-5), (3,3), (5,7)$ .

We will use the two points from the table to find the slope of the second line. Let's use the points $(-3, -9)$ and $(3, 3)$ .
$slope = \frac{y_2 - y_1}{x_2 - x_1}$

Now, plug in the values for $x_1, y_1, x_2, y_2$ to calculate the slope.
$slope = \frac{3 - (-9)}{3 - (-3)}$

Simplify the numerator and denominator.
$slope = \frac{3 + 9}{3 + 3}$
$slope = \frac{12}{6}$

Calculate the slope.
$slope = 2$

Now that we have the slope, we can use the point-slope form to write the equation of the second line. We can use any point from the table; let's use the point $(-3, -9)$ .
$y - y_1 = slope \cdot (x - x_1)$

Plug in the values for $x_1, y_1$ and the slope into the point-slope form.
$y - (-9) = 2 \cdot (x - (-3))$

Simplify the equation.
$y + 9 = 2 \cdot (x + 3)$

Distribute the slope on the right side of the equation.
$y + 9 = 2x + 6$

Subtract 9 from both sides to get the equation in slope-intercept form, $y = mx + b$ .
$y = 2x + 6 - 9$

Simplify the equation.
$y = 2x - 3$

Now we have the second equation of the system. The system of equations is:
$\begin{cases} 3x - 2y = -4 \\ y = 2x - 3 \end{cases}$

To find the solution to the system, we can substitute the second equation into the first equation.
$3x - 2(2x - 3) = -4$

Distribute the $-2$ on the left side of the equation.
$3x - 4x + 6 = -4$

Combine like terms.
$-x + 6 = -4$

Add $x$ to both sides to isolate the $x$ term on one side.
$6 = x - 4$

Add 4 to both sides to solve for $x$ .
$x = 6 + 4$

Calculate the value of $x$ .
$x = 10$

Now that we have the value of $x$ , we can substitute it back into the second equation to find the value of $y$ .
$y = 2(10) - 3$

Calculate the value of $y$ .
$y = 20 - 3$

Simplify to get the final value of $y$ .
$y = 17$
The $y$ -value of the solution to the system is 17.