graph a line through the given point with the given slope.
11. (0,0) and ( m = \frac{3}{5} )
12. (2, -2) and ( m = 4 )
13. (-3, 6) and ( m = -\frac{4}{5} )
14. (-3, -5) and ( m = 0 )
follow the directions.
15. explain why (a) ( y = x - 3 ) and (b) ( \frac{2}{5}x - \frac{1}{2}y = 4 ) are not in standard form.
16. graph (a) ( 3x + 2y = 6 ) and (b) ( -6x + 2y = -12 ) by finding intercepts.
17. the number 3 is a distance of 7 from two numbers. write and solve an equation to find the numbers.
18. jennifer has a plastic bottle filled with 264 coins (pennies and nickels) that have a total value of $4.36. write and solve an equation to find how many of each coin she has.

Question

Accepted Answer

### Problem 11: Graph a line through (0,0) with $ m = \frac{3}{5} $
# Explanation:
## Step 1: Recall Point - Slope Form
The point - slope form of a line is $ y - y_1=m(x - x_1) $, where $ (x_1,y_1) $ is a point on the line and $ m $ is the slope. Here, $ x_1 = 0 $, $ y_1=0 $ and $ m=\frac{3}{5} $. Substituting these values into the point - slope form, we get $ y-0=\frac{3}{5}(x - 0) $, which simplifies to $ y=\frac{3}{5}x $.
## Step 2: Identify the Slope - Intercept Form
The slope - intercept form of a line is $ y=mx + b $, where $ m $ is the slope and $ b $ is the y - intercept. For $ y=\frac{3}{5}x $, the y - intercept $ b = 0 $ (the line passes through the origin) and the slope $ m=\frac{3}{5} $. The slope $ \frac{3}{5} $ means that for every 5 units we move to the right (increase in $ x $ by 5), we move up 3 units (increase in $ y $ by 3).
## Step 3: Plot the Line
1. Start at the point $ (0,0) $ (the origin).
2. Use the slope to find another point. Since the slope $ m=\frac{\text{rise}}{\text{run}}=\frac{3}{5} $, from $ (0,0) $, move 5 units to the right (to $ x = 5 $) and 3 units up (to $ y=3 $). So the point $ (5,3) $ is also on the line.
3. Draw a straight line through the points $ (0,0) $ and $ (5,3) $.

# Answer:
The line $ y = \frac{3}{5}x $ is graphed by plotting the points $ (0,0) $ and $ (5,3) $ (and other points found using the slope) and drawing a straight line through them.

### Problem 12: Graph a line through $ (2,-2) $ with $ m = 4 $
# Explanation:
## Step 1: Use Point - Slope Form
The point - slope form is $ y - y_1=m(x - x_1) $. Here, $ x_1 = 2 $, $ y_1=-2 $ and $ m = 4 $. Substituting these values, we have $ y-(-2)=4(x - 2) $.
Simplify the equation: $ y + 2=4x-8 $, then $ y=4x-8 - 2=4x-10 $.
## Step 2: Identify Key Points
- **Y - intercept**: For the equation $ y = 4x-10 $, when $ x = 0 $, $ y=-10 $. So the y - intercept is $ (0,-10) $.
- **Another point using slope**: The slope $ m = 4=\frac{4}{1} $. From the point $ (2,-2) $, move 1 unit to the right (increase $ x $ by 1: $ x=2 + 1=3 $) and 4 units up (increase $ y $ by 4: $ y=-2+4 = 2 $). So the point $ (3,2) $ is on the line.
## Step 3: Plot the Line
1. Plot the points $ (2,-2) $, $ (0,-10) $ and $ (3,2) $.
2. Draw a straight line through these points.

# Answer:
The line $ y = 4x-10 $ is graphed by plotting the points $ (2,-2) $, $ (0,-10) $ and $ (3,2) $ (and other points) and drawing a straight line through them.

### Problem 13: Graph a line through $ (-3,6) $ with $ m=-\frac{4}{5} $
# Explanation:
## Step 1: Apply Point - Slope Form
The point - slope form is $ y - y_1=m(x - x_1) $. Here, $ x_1=-3 $, $ y_1 = 6 $ and $ m=-\frac{4}{5} $. Substituting, we get $ y - 6=-\frac{4}{5}(x+3) $.
Expand the right - hand side: $ y-6=-\frac{4}{5}x-\frac{12}{5} $.
Add 6 to both sides: $ y=-\frac{4}{5}x-\frac{12}{5}+6 $. Convert 6 to fifths: $ 6=\frac{30}{5} $. Then $ y=-\frac{4}{5}x+\frac{30 - 12}{5}=-\frac{4}{5}x+\frac{18}{5} $.
## Step 2: Find Key Points
- **Y - intercept**: When $ x = 0 $, $ y=\frac{18}{5}=3.6 $. So the y - intercept is $ (0,3.6) $.
- **Using slope from $ (-3,6) $**: The slope $ m=-\frac{4}{5}=\frac{-4}{5} $. From the point $ (-3,6) $, move 5 units to the right (increase $ x $ by 5: $ x=-3 + 5 = 2 $) and 4 units down (decrease $ y $ by 4: $ y=6-4 = 2 $). So the point $ (2,2) $ is on the line.
## Step 3: Plot the Line
1. Plot the points $ (-3,6) $, $ (0,3.6) $ and $ (2,2) $.
2. Draw a straight line through these points.

# Answer:
The line $ y=-\frac{4}{5}x+\frac{18}{5} $ is graphed by plotting the points $ (-3,6) $, $ (0,3.6) $ and $ (2,2) $ (and other points) and drawing a straight line through them.

### Problem 14: Graph a line through $ (-3,-5) $ with $ m = 0 $
# Explanation:
## Step 1: Analyze the Slope
A slope of $ m = 0 $ means that the line is horizontal. The equation of a horizontal line is of the form $ y = k $, where $ k $ is the y - coordinate of any point on the line.
## Step 2: Find the Equation of the Line
Since the line passes through the point $ (-3,-5) $, the y - coordinate of this point is $ - 5 $. So the equation of the line is $ y=-5 $.
## Step 3: Plot the Line
1. The line $ y=-5 $ is a horizontal line. For any value of $ x $, $ y=-5 $.
2. Plot points such as $ (-3,-5) $, $ (0,-5) $, $ (5,-5) $ and draw a straight horizontal line through them.

# Answer:
The line $ y=-5 $ is graphed as a horizontal line passing through points like $ (-3,-5) $, $ (0,-5) $, $ (5,-5) $ etc.

### Problem 15: Explain why (a) $ y=x - 3 $ and (b) $ \frac{2}{5}x-\frac{1}{2}y = 4 $ are not in standard form
# Explanation:
## Step 1: Recall Standard Form of a Line
The standard form of a linear equation is $ Ax+By = C $, where $ A $, $ B $, and $ C $ are integers, and $ A\geq0 $, and $ A $ and $ B $ are not both zero.
## Step 2: Analyze Equation (a) $ y=x - 3 $
Rewrite the equation $ y=x - 3 $ in the form $ Ax+By = C $. Subtract $ x $ and add 3 to both sides: $ -x+y=- 3 $. Here, $ A=-1<0 $, which does not satisfy the condition $ A\geq0 $ for standard form. Also, the original equation is in slope - intercept form ($ y=mx + b $) rather than $ Ax + By=C $ with integer coefficients in the correct form.
## Step 3: Analyze Equation (b) $ \frac{2}{5}x-\frac{1}{2}y = 4 $
The coefficients of $ x $ and $ y $ are fractions ($ \frac{2}{5} $ and $ -\frac{1}{2} $). In standard form $ Ax+By = C $, $ A $, $ B $, and $ C $ should be integers. To eliminate the fractions, we can multiply the entire equation by the least common multiple of 5 and 2, which is 10. Multiplying gives $ 10\times\frac{2}{5}x-10\times\frac{1}{2}y=10\times4 $, or $ 4x - 5y = 40 $ (which is in standard form), but the original equation has fractional coefficients, so it is not in standard form.

# Answer:
(a) $ y=x - 3 $ is not in standard form ($ Ax + By=C $, $ A\geq0 $, $ A,B,C\in\mathbb{Z}$) because when rewritten as $ -x + y=-3 $, $ A=-1<0 $ and it is in slope - intercept form. (b) $ \frac{2}{5}x-\frac{1}{2}y = 4 $ is not in standard form because it has fractional coefficients for $ x $ and $ y $, while standard form requires integer coefficients for $ A $, $ B $, and $ C $.

### Problem 16: Graph (a) $ 3x + 2y=6 $ and (b) $ - 6x+2y=-12 $ by finding intercepts
# Explanation:
#### Part (a): Graph $ 3x + 2y=6 $
## Step 1: Find the x - intercept
The x - intercept is the point where $ y = 0 $. Substitute $ y = 0 $ into the equation $ 3x+2y = 6 $:
$ 3x+2(0)=6 $
$ 3x=6 $
Divide both sides by 3: $ x = 2 $. So the x - intercept is $ (2,0) $.
## Step 2: Find the y - intercept
The y - intercept is the point where $ x = 0 $. Substitute $ x = 0 $ into the equation $ 3x+2y = 6 $:
$ 3(0)+2y=6 $
$ 2y=6 $
Divide both sides by 2: $ y = 3 $. So the y - intercept is $ (0,3) $.
## Step 3: Plot the Line
Plot the points $ (2,0) $ and $ (0,3) $ and draw a straight line through them.

#### Part (b): Graph $ -6x + 2y=-12 $
## Step 1: Find the x - intercept
Substitute $ y = 0 $ into the equation $ -6x+2y=-12 $:
$ -6x+2(0)=-12 $
$ -6x=-12 $
Divide both sides by $ - 6 $: $ x = 2 $. So the x - intercept is $ (2,0) $.
## Step 2: Find the y - intercept
Substitute $ x = 0 $ into the equation $ -6x+2y=-12 $:
$ -6(0)+2y=-12 $
$ 2y=-12 $
Divide both sides by 2: $ y=-6 $. So the y - intercept is $ (0,-6) $.
## Step 3: Plot the Line
Plot the points $ (2,0) $ and $ (0,-6) $ and draw a straight line through them.

# Answer:
(a) For $ 3x + 2y = 6 $, x - intercept $ (2,0) $, y - intercept $ (0,3) $, line through these points. (b) For $ -6x + 2y=-12 $, x - intercept $ (2,0) $, y - intercept $ (0,-6) $, line through these points.

### Problem 17: The number 3 is a distance of 7 from two numbers. Find the numbers
# Explanation:
## Step 1: Set Up the Absolute Value Equation
Let the number be $ x $. The distance between $ x $ and 3 is given by $ |x - 3| $. We know that this distance is 7. So we set up the equation $ |x - 3|=7 $.
## Step 2: Solve the Absolute Value Equation
The absolute value equation $ |a|=b $ (where $ b>0 $) has two solutions: $ a = b $ or $ a=-b $.
For $ |x - 3|=7 $, we have two cases:
- Case 1: $ x - 3=7 $
Add 3 to both sides: $ x=7 + 3=10 $
- Case 2: $ x - 3=-7 $
Add 3 to both sides: $ x=-7 + 3=-4 $

# Answer:
The two numbers are $ - 4 $ and $ 10 $.

### Problem 18: Jennifer has 264 coins (pennies and nickels) worth $4.36. Find the number of each coin
# Explanation:
## Step 1: Define Variables
Let $ p $ be the number of pennies and $ n $ be the number of nickels.
## Step 2: Set Up Equations
- **Equation 1 (Total number of coins)**: The total number of coins is 264, so $ p + n=264 $.
- **Equation 2 (Total value of coins)**: The value of a penny is $0.01 and the value of a nickel is $0.05. The total value of the coins is $4.36. So $ 0.01p+0.05n = 4.36 $.
## Step 3: Solve the System of Equations
From Equation 1, we can express $ p=264 - n $.
Substitute $ p = 264 - n $ into Equation 2:
$ 0.01(264 - n)+0.05n=4.36 $
Expand the left - hand side: $ 2.64-0.01n + 0.05n=4.36 $
Combine like terms: $ 2.64 + 0.04n=4.36 $
Subtract 2.64 from both sides: $ 0.04n=4.36 - 2.64=1.72 $
Divide both sides by 0.04: $ n=\frac{1.72}{0.04}=43 $
## Step 4: Find the Number of Pennies
Substitute $ n = 43 $ into $ p=264 - n $: $ p=264 - 43 = 221 $

# Answer:
The number of pennies is 221 and the number of nickels is 43.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 11: Graph a line through (0,0) with \( m = \frac{3}{5} \)

Step 1: Recall Point - Slope Form

Step 2: Identify the Slope - Intercept Form

Step 3: Plot the Line

Step 1: Use Point - Slope Form

Step 2: Identify Key Points

Step 3: Plot the Line

Step 1: Apply Point - Slope Form

Step 2: Find Key Points

Step 3: Plot the Line

Answer:

Problem 12: Graph a line through \( (2,-2) \) with \( m = 4 \)