Solution:
Graph #2

Solution:
Graph #1

Solution:
Graph #3
Details:
The equation, \(y={\color{Red}4}x−{\color{Blue}3}\), is written in slope-intercept form, which is \(y = {\color{Red}m}x + {\color{Blue}b}\), where m is the slope and b is the y-intercept. We need to find the graph that has a \({\color{Red}slope}\) of \({\color{Red}4}\) and a \({\color{Blue}y-intercept}\) of \({\color{Blue}-3}\).

First, we look at Graph #1. We see that the y-intercept is 3 and the slope is negative. It is not the correct graph.



Then we look at Graph #2. We see that the y-intercept is \(−3\) and the slope is positive. The rise is 4 and the run is 1, so the slope is \(m=\dfrac{4}{1}\), which is equal to 4. This is the correct graph.



We can also check Graph #3. We see that the y-intercept is 0 and the slope is positive. The rise is 1 and the run is 3 so the slope is \(m=\dfrac{1}{3}\).This is not the correct graph.

Solution:
Note: Your line may look different than this one, but the x-intercept and the y-intercept should be the same.

Solution:
Note: Your line may look different than this one, but the x-intercept and the y-intercept should be the same.

Solution:
Note: Your line may look different than this one, but the x-intercept and the y-intercept should be the same.

Solution:
Note: Your line may look different than this one, but the x-intercept and the y-intercept should be the same.

Details:
To graph the line, \(\dfrac{1}{3}{\text{x}}{-}{\dfrac{1}{2}}{\text{y}}=5\) , we need to find at least two points that are on the line. We’ll start with a table of values:

x	y
-	-
-	-

To find the y-intercept, we’ll plug in 0 for x and find the corresponding y-value:

\({\dfrac{1}{3}(0)}{-}{\dfrac{1}{2}}{\text{y}}=5\)

We know that \(\dfrac{1}{3}\times0=0\) so our equation now reads:

\(0-\dfrac{1}{2}{\text{y}}=5\)

Which equals:

\(-\dfrac{1}{2}{\text{y}}=5\)

Next, we multiply both sides by \(-\dfrac{2}{1}\) to cancel out the \(-\dfrac{1}{2}\) and solve for y. We also write 5 as \(\frac{5}{1}\) to make solving easier:

\(\left(-\dfrac{2}{1}\right)\left(-\dfrac{1}{2}{\text{y}}\right)=\dfrac{5}{1}\left(-\dfrac{2}{1}\right)\)

On the left side we multiply \(\left(-\dfrac{2}{1}\right)\times\left(-\dfrac{1}{2}\right)\) together which equals 1. We now have:

\(1{\text{y}}=\dfrac{5}{1}\left(-\dfrac{2}{1}\right)\)

To multiply \(\dfrac{5}{1}\times\left(-\dfrac{2}{1}\right)\) , we multiply straight across which equals \(-\frac{10}{1}\).

\(1{\text{y}}= -\dfrac{10}{1}\)

Which simplifies to:

\(1{\text{y}}=-10\)

So the point \((0,−10)\) is on the line and we can update our table accordingly.

x	y
0	-10
-	-

To find the x-intercept, we follow the same process plugging in 0 for y and then finding the corresponding x-value:

\(\dfrac{1}{3}{\text{x}}{-}{\dfrac{1}{2}}\left(0\right)=5\)

Which equals:

\(\dfrac{1}{3}{\text{x}}{-}0=5\)

Which equals:

\(\frac{1}{3}{\text{x}}=5\)

Then multiply both sides by 3:

\((3)(\dfrac{1}{3}){\text{x}}=5(3)\)

Which gives us:

\((\cancel{3})(\dfrac{1}{\cancel{3}}){\text{x}}=5(3)\)

and

\({\text{x}}=15\)

Now we have an additional point, (15,0).

x	y
0	-10
15	0

Using the points \((15,0)\) and \((0,−10)\) we can graph the line. First, we graph the points:



Then we draw a line through both points:

Solution:

Details:
The equation of our line is \(y=−3x+{\color{Red}2}\) and it is written in slope-intercept form, which is \(y=mx + {\color{Red}b}\), where m is the slope and \({\color{Red}b}\) is the \({\color{Red}y-intercept}\). So the y-intercept is (0,2)

Solution:
Note: Answers will vary depending on what value of x you choose. Several possible points are shown. All correct points will lie upon the line, but since the line is infinite we cannot show the entire line or all possible points on it.

Details:
We want to graph this line and already have the y-intercept which is (0, 2). We need one more point to graph the line. We’ll choose \({\color{Red}x = 1}\) and find the corresponding y-value (you can choose any value for x. We chose x = 1 for convenience). Our equation is:

\(y=−3{\color{Red}x}+2\)

Replace x with \({\color{Red}1}\):

\(y=−3{\color{Red}(1)}+2\)

Then simplify:

\(y=−{\color{Red}3}+2\)

Then add which gives us:

\(y=−1\)

So another point on our line is \((1,−1)\). Now we can graph both points:



Then draw a line through both points to graph the line:

Solution:
Note: Answers will vary depending on what value of x you choose. Several possible points are shown. All correct points will lie upon the line, but since the line is infinite we cannot show the entire line or all possible points on it.

Solution:
Note: Answers will vary depending on what value of x you choose. Several possible points are shown. All correct points will lie upon the line, but since the line is infinite we cannot show the entire line or all possible points on it.