Math / Algebra

QuestionGraph the equation. If necessary, write the equation in slope-intercept form first.
10. $y=x+1$
11. $y=x-6$
13. $y=-2 x$
14. $y=2 x$
16. $y=4$
17. $y=\frac{1}{2} x-1$
19. $y=\frac{3}{2} x+\frac{1}{2}$
20. $-3 x+y=8$

Studdy Solution

STEP 1

What is this asking? We need to graph a bunch of equations, and if they're not already in slope-intercept form ( $y = mx + b$ ), we need to rewrite them first! Watch out! Remember, the slope ( $m$ ) tells us how steep the line is, and the y-intercept ( $b$ ) tells us where the line crosses the y-axis.
Don't mix them up!

STEP 2

1. Graph $y = x + 1$
2. Graph $y = x - 6$
3. Graph $y = -2x$
4. Graph $y = 2x$
5. Graph $y = 4$
6. Graph $y = \frac{1}{2}x - 1$
7. Graph $y = \frac{3}{2}x + \frac{1}{2}$
8. Graph $-3x + y = 8$

STEP 3

This equation is already in slope-intercept form!
The slope is $m = 1$ (think of it as $\frac{1}{1}$ ), and the y-intercept is $b = 1$ .

STEP 4

To graph it, start by plotting the y-intercept $(0, 1)$ .
From there, use the slope to find another point.
A slope of $\frac{1}{1}$ means we go up 1 unit and to the right 1 unit.
This gives us the point $(1, 2)$ .

STEP 5

Draw a line through the points $(0, 1)$ and $(1, 2)$ , and there you have it!

STEP 6

Slope-intercept form?
Check! Here, $m = 1$ and $b = -6$ .

STEP 7

Plot the y-intercept $(0, -6)$ .
Then, using the slope of $1$ (or $\frac{1}{1}$ ), go up 1 and right 1 to find the point $(1, -5)$ .

STEP 8

Connect the dots $(0, -6)$ and $(1, -5)$ with a straight line.
Boom!

STEP 9

We're in slope-intercept form, with $m = -2$ (think $\frac{-2}{1}$ ) and $b = 0$ .

STEP 10

The y-intercept is $(0, 0)$ , the origin!
With a slope of $\frac{-2}{1}$ , go down 2 units and right 1 unit to find the point $(1, -2)$ .

STEP 11

Draw a line through $(0, 0)$ and $(1, -2)$ .
Perfect!

STEP 12

Slope-intercept form, with $m = 2$ (or $\frac{2}{1}$ ) and $b = 0$ .

STEP 13

The y-intercept is at the origin, $(0, 0)$ .
Using the slope, go up 2 and right 1 to the point $(1, 2)$ .

STEP 14

Draw a line through $(0, 0)$ and $(1, 2)$ !

STEP 15

This is a horizontal line!
Think of it as $y = 0x + 4$ , so $m = 0$ and $b = 4$ .

STEP 16

Plot the y-intercept $(0, 4)$ .
Since the slope is 0, the line is perfectly flat, going through all points with a y-coordinate of 4.

STEP 17

Slope-intercept form, with $m = \frac{1}{2}$ and $b = -1$ .

STEP 18

Plot the y-intercept $(0, -1)$ .
Using the slope, go up 1 and right 2 to $(2, 0)$ .

STEP 19

Connect $(0, -1)$ and $(2, 0)$ with a line!

STEP 20

Here, $m = \frac{3}{2}$ and $b = \frac{1}{2}$ .

STEP 21

Plot the y-intercept $(0, \frac{1}{2})$ .
Using the slope, go up 3 and right 2 to $(2, \frac{7}{2})$ or $(2, 3.5)$ .

STEP 22

Draw the line through $(0, \frac{1}{2})$ and $(2, \frac{7}{2})$ .

STEP 23

Not in slope-intercept form!
Let's fix that.
We want to isolate $y$ .

STEP 24

Add $3x$ to both sides of the equation: $-3x + y + 3x = 8 + 3x$ .
This simplifies to $y = 3x + 8$ .
Now we have $m = 3$ and $b = 8$ .

STEP 25

Plot the y-intercept $(0, 8)$ .
Use the slope to go up 3 and right 1 to $(1, 11)$ .

STEP 26

Draw a line through $(0, 8)$ and $(1, 11)$ , and you're done!

STEP 27

We've graphed all the equations!
Each line is beautifully displayed, showcasing its unique slope and y-intercept.

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Studdy Solution

STEP 1

STEP 2

1. Graph y=x+1y = x + 1y=x+12. Graph y=x−6y = x - 6y=x−63. Graph y=−2xy = -2xy=−2x4. Graph y=2xy = 2xy=2x5. Graph y=4y = 4y=46. Graph y=12x−1y = \frac{1}{2}x - 1y=21​x−17. Graph y=32x+12y = \frac{3}{2}x + \frac{1}{2}y=23​x+21​8. Graph −3x+y=8-3x + y = 8−3x+y=8

STEP 3

This equation is already in slope-intercept form!The **slope** is m=1m = 1m=1 (think of it as 11\frac{1}{1}11​), and the **y-intercept** is b=1b = 1b=1.

STEP 4

To graph it, start by plotting the **y-intercept** (0,1)(0, 1)(0,1).From there, use the **slope** to find another point.A slope of 11\frac{1}{1}11​ means we go up **1** unit and to the right **1** unit.This gives us the point (1,2)(1, 2)(1,2).

STEP 5

Draw a line through the points (0,1)(0, 1)(0,1) and (1,2)(1, 2)(1,2), and there you have it!

STEP 6

Slope-intercept form?Check! Here, m=1m = 1m=1 and b=−6b = -6b=−6.

STEP 7

Plot the **y-intercept** (0,−6)(0, -6)(0,−6).Then, using the **slope** of 111 (or 11\frac{1}{1}11​), go up **1** and right **1** to find the point (1,−5)(1, -5)(1,−5).

STEP 8

Connect the dots (0,−6)(0, -6)(0,−6) and (1,−5)(1, -5)(1,−5) with a straight line.Boom!

STEP 9

We're in slope-intercept form, with m=−2m = -2m=−2 (think −21\frac{-2}{1}1−2​) and b=0b = 0b=0.

STEP 10

The **y-intercept** is (0,0)(0, 0)(0,0), the origin!With a slope of −21\frac{-2}{1}1−2​, go down **2** units and right **1** unit to find the point (1,−2)(1, -2)(1,−2).

STEP 11

Draw a line through (0,0)(0, 0)(0,0) and (1,−2)(1, -2)(1,−2).Perfect!

STEP 12

Slope-intercept form, with m=2m = 2m=2 (or 21\frac{2}{1}12​) and b=0b = 0b=0.

STEP 13

The **y-intercept** is at the origin, (0,0)(0, 0)(0,0).Using the **slope**, go up **2** and right **1** to the point (1,2)(1, 2)(1,2).

STEP 14

Draw a line through (0,0)(0, 0)(0,0) and (1,2)(1, 2)(1,2)!

STEP 15

This is a horizontal line!Think of it as y=0x+4y = 0x + 4y=0x+4, so m=0m = 0m=0 and b=4b = 4b=4.

STEP 16

Plot the **y-intercept** (0,4)(0, 4)(0,4).Since the slope is **0**, the line is perfectly flat, going through all points with a y-coordinate of **4**.

STEP 17

Slope-intercept form, with m=12m = \frac{1}{2}m=21​ and b=−1b = -1b=−1.

STEP 18

Plot the **y-intercept** (0,−1)(0, -1)(0,−1).Using the **slope**, go up **1** and right **2** to (2,0)(2, 0)(2,0).

STEP 19

Connect (0,−1)(0, -1)(0,−1) and (2,0)(2, 0)(2,0) with a line!

STEP 20

Here, m=32m = \frac{3}{2}m=23​ and b=12b = \frac{1}{2}b=21​.

STEP 21

Plot the **y-intercept** (0,12)(0, \frac{1}{2})(0,21​).Using the **slope**, go up **3** and right **2** to (2,72)(2, \frac{7}{2})(2,27​) or (2,3.5)(2, 3.5)(2,3.5).

STEP 22

Draw the line through (0,12)(0, \frac{1}{2})(0,21​) and (2,72)(2, \frac{7}{2})(2,27​).

STEP 23

Not in slope-intercept form!Let's fix that.We want to isolate yyy.

STEP 24

Add 3x3x3x to both sides of the equation: −3x+y+3x=8+3x-3x + y + 3x = 8 + 3x−3x+y+3x=8+3x.This simplifies to y=3x+8y = 3x + 8y=3x+8.Now we have m=3m = 3m=3 and b=8b = 8b=8.

STEP 25

Plot the **y-intercept** (0,8)(0, 8)(0,8).Use the **slope** to go up **3** and right **1** to (1,11)(1, 11)(1,11).

STEP 26

Draw a line through (0,8)(0, 8)(0,8) and (1,11)(1, 11)(1,11), and you're done!

STEP 27

We've graphed all the equations!Each line is beautifully displayed, showcasing its unique slope and y-intercept.

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1. Graph $y = x + 1$
2. Graph $y = x - 6$
3. Graph $y = -2x$
4. Graph $y = 2x$
5. Graph $y = 4$
6. Graph $y = \frac{1}{2}x - 1$
7. Graph $y = \frac{3}{2}x + \frac{1}{2}$
8. Graph $-3x + y = 8$

This equation is already in slope-intercept form!
The slope is $m = 1$ (think of it as $\frac{1}{1}$ ), and the y-intercept is $b = 1$ .

To graph it, start by plotting the y-intercept $(0, 1)$ .
From there, use the slope to find another point.
A slope of $\frac{1}{1}$ means we go up 1 unit and to the right 1 unit.
This gives us the point $(1, 2)$ .

Draw a line through the points $(0, 1)$ and $(1, 2)$ , and there you have it!

Slope-intercept form?
Check! Here, $m = 1$ and $b = -6$ .

Plot the y-intercept $(0, -6)$ .
Then, using the slope of $1$ (or $\frac{1}{1}$ ), go up 1 and right 1 to find the point $(1, -5)$ .

Connect the dots $(0, -6)$ and $(1, -5)$ with a straight line.
Boom!

We're in slope-intercept form, with $m = -2$ (think $\frac{-2}{1}$ ) and $b = 0$ .

The y-intercept is $(0, 0)$ , the origin!
With a slope of $\frac{-2}{1}$ , go down 2 units and right 1 unit to find the point $(1, -2)$ .

Draw a line through $(0, 0)$ and $(1, -2)$ .
Perfect!

Slope-intercept form, with $m = 2$ (or $\frac{2}{1}$ ) and $b = 0$ .

The y-intercept is at the origin, $(0, 0)$ .
Using the slope, go up 2 and right 1 to the point $(1, 2)$ .

Draw a line through $(0, 0)$ and $(1, 2)$ !

This is a horizontal line!
Think of it as $y = 0x + 4$ , so $m = 0$ and $b = 4$ .

Plot the y-intercept $(0, 4)$ .
Since the slope is 0, the line is perfectly flat, going through all points with a y-coordinate of 4.

Slope-intercept form, with $m = \frac{1}{2}$ and $b = -1$ .

Plot the y-intercept $(0, -1)$ .
Using the slope, go up 1 and right 2 to $(2, 0)$ .

Connect $(0, -1)$ and $(2, 0)$ with a line!

Here, $m = \frac{3}{2}$ and $b = \frac{1}{2}$ .

Plot the y-intercept $(0, \frac{1}{2})$ .
Using the slope, go up 3 and right 2 to $(2, \frac{7}{2})$ or $(2, 3.5)$ .

Draw the line through $(0, \frac{1}{2})$ and $(2, \frac{7}{2})$ .

Not in slope-intercept form!
Let's fix that.
We want to isolate $y$ .

Add $3x$ to both sides of the equation: $-3x + y + 3x = 8 + 3x$ .
This simplifies to $y = 3x + 8$ .
Now we have $m = 3$ and $b = 8$ .

Plot the y-intercept $(0, 8)$ .
Use the slope to go up 3 and right 1 to $(1, 11)$ .

Draw a line through $(0, 8)$ and $(1, 11)$ , and you're done!

We've graphed all the equations!
Each line is beautifully displayed, showcasing its unique slope and y-intercept.