Math / Algebra

Question2. The graph of a linear function 9 passes through the points $(-7,-4)$ and $(7,6)$ .
What are the slope and $y$ -intercept of the graph of $g$ ?
4. The slope is $\frac{7}{5}$ and the $y$ -intercept is -1 . B. The slope is $\frac{7}{5}$ and the $y$ -intercept is 1 . C. The slope is $\frac{5}{7}$ and the $y$ -intercept is -1 . D. The slope is $\frac{5}{7}$ and the $y$ -intercept is 1 .
3. Which ordered pair is in the solution set of $y \leq \frac{3}{5} x-6$ ? A. $(5,-4)$ B. $(-2,-5)$ C. $(9,1)$ D. $(-8,3)$

Studdy Solution

STEP 1

What is this asking? We're given two points on a line and need to find the line's slope and where it crosses the y-axis.
Then, we need to figure out which point satisfies a given inequality. Watch out! Don't mix up the x and y coordinates when calculating the slope!
Also, remember that the y-intercept is where the line crosses the y-axis, meaning $x = 0$ .
For the inequality, make sure to test all the points, and don't forget that "less than or equal to" means the point on the line is also a solution!

STEP 2

1. Find the Slope
2. Find the Y-Intercept
3. Check Inequality Solutions

STEP 3

Alright, let's find the slope!
Remember, the slope is the "rise over run," how much the y-value changes for every change in x.
We have our two points, $(-7, -4)$ and $(7, 6)$ .

STEP 4

So, our "rise" is the change in y: $6 - (-4) = 6 + 4 = \mathbf{10}$ .
Our "run" is the change in x: $7 - (-7) = 7 + 7 = \mathbf{14}$ .

STEP 5

Therefore, our slope is $\frac{\text{rise}}{\text{run}} = \frac{10}{14}$ .
We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2.
So, $\frac{10}{14} = \frac{10 \div 2}{14 \div 2} = \frac{5}{7}$ .
Awesome!

STEP 6

Now, let's find the y-intercept!
We know the equation of a line is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
We already found our slope, $m = \frac{5}{7}$ .

STEP 7

We can plug in one of our points and our slope to solve for $b$ .
Let's use the point $(7, 6)$ .
So, we have $6 = \frac{5}{7} \cdot 7 + b$ .

STEP 8

Multiplying $\frac{5}{7}$ by 7 gives us 5 (since $\frac{5}{7} \cdot 7 = \frac{5 \cdot 7}{7} = \frac{5}{1} = 5$ ).
So, our equation becomes $6 = 5 + b$ .

STEP 9

To isolate $b$ , we subtract 5 from both sides of the equation: $6 - 5 = 5 + b - 5$ , which simplifies to $1 = b$ .
So, our y-intercept is $\mathbf{1}$ !

STEP 10

Now, let's check which ordered pair satisfies the inequality $y \leq \frac{3}{5}x - 6$ .
We'll test each point one by one.

STEP 11

For point A $(5, -4)$ : $-4 \leq \frac{3}{5} \cdot 5 - 6$ .
This simplifies to $-4 \leq 3 - 6$ , or $-4 \leq -3$ .
This is true!

STEP 12

For point B $(-2, -5)$ : $-5 \leq \frac{3}{5} \cdot (-2) - 6$ .
This simplifies to $-5 \leq -\frac{6}{5} - 6$ , or $-5 \leq -\frac{36}{5}$ .
Since $-\frac{36}{5}$ is equal to -7.2, we have $-5 \leq -7.2$ .
This is false.

STEP 13

For point C $(9, 1)$ : $1 \leq \frac{3}{5} \cdot 9 - 6$ .
This simplifies to $1 \leq \frac{27}{5} - 6$ , or $1 \leq \frac{27}{5} - \frac{30}{5}$ , which gives us $1 \leq -\frac{3}{5}$ .
This is false.

STEP 14

For point D $(-8, 3)$ : $3 \leq \frac{3}{5} \cdot (-8) - 6$ .
This simplifies to $3 \leq -\frac{24}{5} - 6$ , or $3 \leq -\frac{24}{5} - \frac{30}{5}$ , which gives us $3 \leq -\frac{54}{5}$ .
This is false.

STEP 15

The slope of the line is $\frac{5}{7}$ and the y-intercept is 1 (which corresponds to answer D).
The ordered pair $(5, -4)$ is in the solution set of the inequality.

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Studdy Solution

STEP 1

STEP 2

1. Find the Slope
2. Find the Y-Intercept
3. Check Inequality Solutions

STEP 3

Alright, let's find the slope!
Remember, the slope is the "rise over run," how much the y-value changes for every change in x.
We have our two points, $(-7, -4)$ and $(7, 6)$ .

STEP 4

So, our "rise" is the change in y: $6 - (-4) = 6 + 4 = \mathbf{10}$ .
Our "run" is the change in x: $7 - (-7) = 7 + 7 = \mathbf{14}$ .

STEP 5

Therefore, our slope is $\frac{\text{rise}}{\text{run}} = \frac{10}{14}$ .
We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2.
So, $\frac{10}{14} = \frac{10 \div 2}{14 \div 2} = \frac{5}{7}$ .
Awesome!

STEP 6

Now, let's find the y-intercept!
We know the equation of a line is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
We already found our slope, $m = \frac{5}{7}$ .

STEP 7

We can plug in one of our points and our slope to solve for $b$ .
Let's use the point $(7, 6)$ .
So, we have $6 = \frac{5}{7} \cdot 7 + b$ .

STEP 8

Multiplying $\frac{5}{7}$ by 7 gives us 5 (since $\frac{5}{7} \cdot 7 = \frac{5 \cdot 7}{7} = \frac{5}{1} = 5$ ).
So, our equation becomes $6 = 5 + b$ .

STEP 9

To isolate $b$ , we subtract 5 from both sides of the equation: $6 - 5 = 5 + b - 5$ , which simplifies to $1 = b$ .
So, our y-intercept is $\mathbf{1}$ !

STEP 10

Now, let's check which ordered pair satisfies the inequality $y \leq \frac{3}{5}x - 6$ .
We'll test each point one by one.

STEP 11

For point A $(5, -4)$ : $-4 \leq \frac{3}{5} \cdot 5 - 6$ .
This simplifies to $-4 \leq 3 - 6$ , or $-4 \leq -3$ .
This is true!

STEP 12

For point B $(-2, -5)$ : $-5 \leq \frac{3}{5} \cdot (-2) - 6$ .
This simplifies to $-5 \leq -\frac{6}{5} - 6$ , or $-5 \leq -\frac{36}{5}$ .
Since $-\frac{36}{5}$ is equal to -7.2, we have $-5 \leq -7.2$ .
This is false.

STEP 13

For point C $(9, 1)$ : $1 \leq \frac{3}{5} \cdot 9 - 6$ .
This simplifies to $1 \leq \frac{27}{5} - 6$ , or $1 \leq \frac{27}{5} - \frac{30}{5}$ , which gives us $1 \leq -\frac{3}{5}$ .
This is false.

STEP 14

For point D $(-8, 3)$ : $3 \leq \frac{3}{5} \cdot (-8) - 6$ .
This simplifies to $3 \leq -\frac{24}{5} - 6$ , or $3 \leq -\frac{24}{5} - \frac{30}{5}$ , which gives us $3 \leq -\frac{54}{5}$ .
This is false.

STEP 15

The slope of the line is $\frac{5}{7}$ and the y-intercept is 1 (which corresponds to answer D).
The ordered pair $(5, -4)$ is in the solution set of the inequality.

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Studdy Solution

STEP 1

STEP 2

1. Find the Slope2. Find the Y-Intercept3. Check Inequality Solutions

STEP 3

Alright, let's **find the slope**!Remember, the slope is the "rise over run," how much the y-value changes for every change in x.We have our two points, (−7,−4)(-7, -4)(−7,−4) and (7,6)(7, 6)(7,6).

STEP 4

So, our "rise" is the change in y: 6−(−4)=6+4=106 - (-4) = 6 + 4 = \mathbf{10}6−(−4)=6+4=10.Our "run" is the change in x: 7−(−7)=7+7=147 - (-7) = 7 + 7 = \mathbf{14}7−(−7)=7+7=14.

STEP 5

STEP 6

Now, let's **find the y-intercept**!We know the equation of a line is y=mx+by = mx + by=mx+b, where mmm is the slope and bbb is the y-intercept.We already found our slope, m=57m = \frac{5}{7}m=75​.

STEP 7

We can plug in one of our points and our slope to solve for bbb.Let's use the point (7,6)(7, 6)(7,6).So, we have 6=57⋅7+b6 = \frac{5}{7} \cdot 7 + b6=75​⋅7+b.

STEP 8

Multiplying 57\frac{5}{7}75​ by 7 gives us 5 (since 57⋅7=5⋅77=51=5\frac{5}{7} \cdot 7 = \frac{5 \cdot 7}{7} = \frac{5}{1} = 575​⋅7=75⋅7​=15​=5).So, our equation becomes 6=5+b6 = 5 + b6=5+b.

STEP 9

To isolate bbb, we subtract 5 from both sides of the equation: 6−5=5+b−56 - 5 = 5 + b - 56−5=5+b−5, which simplifies to 1=b1 = b1=b.So, our **y-intercept** is 1\mathbf{1}1!

STEP 10

Now, let's **check which ordered pair** satisfies the inequality y≤35x−6y \leq \frac{3}{5}x - 6y≤53​x−6.We'll test each point one by one.

STEP 11

For point A (5,−4)(5, -4)(5,−4): −4≤35⋅5−6-4 \leq \frac{3}{5} \cdot 5 - 6−4≤53​⋅5−6.This simplifies to −4≤3−6-4 \leq 3 - 6−4≤3−6, or −4≤−3-4 \leq -3−4≤−3.This is **true**!

STEP 12

STEP 13

STEP 14

STEP 15

The slope of the line is 57\frac{5}{7}75​ and the y-intercept is 1 (which corresponds to answer D).The ordered pair (5,−4)(5, -4)(5,−4) is in the solution set of the inequality.

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Start learning now

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1. Find the Slope
2. Find the Y-Intercept
3. Check Inequality Solutions

Alright, let's find the slope!
Remember, the slope is the "rise over run," how much the y-value changes for every change in x.
We have our two points, $(-7, -4)$ and $(7, 6)$ .

So, our "rise" is the change in y: $6 - (-4) = 6 + 4 = \mathbf{10}$ .
Our "run" is the change in x: $7 - (-7) = 7 + 7 = \mathbf{14}$ .

Now, let's find the y-intercept!
We know the equation of a line is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
We already found our slope, $m = \frac{5}{7}$ .

We can plug in one of our points and our slope to solve for $b$ .
Let's use the point $(7, 6)$ .
So, we have $6 = \frac{5}{7} \cdot 7 + b$ .

Multiplying $\frac{5}{7}$ by 7 gives us 5 (since $\frac{5}{7} \cdot 7 = \frac{5 \cdot 7}{7} = \frac{5}{1} = 5$ ).
So, our equation becomes $6 = 5 + b$ .

To isolate $b$ , we subtract 5 from both sides of the equation: $6 - 5 = 5 + b - 5$ , which simplifies to $1 = b$ .
So, our y-intercept is $\mathbf{1}$ !

Now, let's check which ordered pair satisfies the inequality $y \leq \frac{3}{5}x - 6$ .
We'll test each point one by one.

For point A $(5, -4)$ : $-4 \leq \frac{3}{5} \cdot 5 - 6$ .
This simplifies to $-4 \leq 3 - 6$ , or $-4 \leq -3$ .
This is true!

The slope of the line is $\frac{5}{7}$ and the y-intercept is 1 (which corresponds to answer D).
The ordered pair $(5, -4)$ is in the solution set of the inequality.