Given the equation
�
=
4
�
y=4x fill in the missing x-value and y-value.
(
3
,
�
)
�
=
(3, y) y=
(
�
,
10
)
�
=
(x, 10) x=
In the equation y = 4x, we need to find the missing x and y values.
1) For the coordinates (3, ?):
Substitute x = 3 in y = 4x:
y = 4(3)
y = 12
So, the missing y-value is 12. Therefore, the coordinates are (3, 12).
2) For the coordinates (?, 10):
Substitute y = 10 in y = 4x:
10 = 4x
Divide both sides by 4:
10/4 = x
x = 2.5
So, the missing x-value is 2.5. Therefore, the coordinates are (2.5, 10).
he distance traveled by the red car is represented by the graph below. Click Here for Help Video.
The distance traveled by the blue car was monitored by a table. Click Here for Help Video.
Number of Hours
Miles Traveled
3
3
162
162
4
4
216
216
5
5
270
270
The speed of the red car is
miles per hour.
The speed of the blue car is
miles per hour.
The
car is faster.
To find the speed of the red car, we need to find the slope of the line representing the distance traveled. Looking at the graph, we can see that the red car traveled a distance of 216 miles in 4 hours.
Slope = Change in distance / Change in time
= (216 - 0) / (4 - 0)
= 216 / 4
= 54 miles per hour
So, the speed of the red car is 54 miles per hour.
To find the speed of the blue car, we can use the table provided. The table shows the number of hours and the miles traveled by the blue car. The blue car traveled a distance of 270 miles in 5 hours.
Slope = Change in distance / Change in time
= (270 - 0) / (5 - 0)
= 270 / 5
= 54 miles per hour
So, the speed of the blue car is also 54 miles per hour.
Since both cars have the same speed, neither car is faster.
Transform
9
�
+
−
4
�
=
−
6
9x+−4y=−6 into slope-intercept form (
�
=
�
�
+
�
y=mx+b ).
To transform the equation 9x + -4y = -6 into slope-intercept form (y = mx + b):
Step 1: Move the term with x to the other side of the equation:
-4y = -9x - 6
Step 2: Divide both sides of the equation by -4 to isolate y:
y = (-9x - 6)/-4
Simplifying the right side:
y = (9/4)x + 3/2
So, the equation 9x + -4y = -6 in slope-intercept form is y = (9/4)x + 3/2.
What is the unit rate if
7
7 bags of chip cost a total of
$
40.37
$40.37? Round your answers to the nearest hundredth.
The unit rate for bags of chips per dollar is
bags per $.
The unit rate of dollars per bag of chips is $
per bag.
To find the unit rate, we need to divide the total cost by the number of bags.
The unit rate for bags of chips per dollar is:
7 bags / $40.37 ≈ 0.173 bags per dollar
The unit rate of dollars per bag of chips is:
$40.37 / 7 bags ≈ $5.77 per bag
The distance traveled by the blue car is represented by
�
=
25
�
y=25x .
The distance traveled by the red car was monitored by a table.
Number of Hours
Miles Traveled
4
4
108
108
5
5
135
135
6
6
162
162
The speed of the blue car is
miles per hour.
The speed of the red car is
miles per hour.
The
car is faster.
To find the speed of the blue car, we can look at the equation y = 25x. Since the equation is in the form y = mx, where m represents the slope, the slope of this equation represents the speed.
In this case, the slope is 25. So, the speed of the blue car is 25 miles per hour.
To find the speed of the red car, we can use the table provided. The table shows the number of hours and the miles traveled by the red car.
For the red car:
- The distance traveled from 4 hours to 5 hours is 135 - 108 = 27 miles.
- The distance traveled from 5 hours to 6 hours is 162 - 135 = 27 miles.
We can see that the red car traveled a consistent distance in each hour, indicating a constant speed. Therefore, the speed of the red car is 27 miles per hour.
Since 27 miles per hour is greater than 25 miles per hour, the red car is faster.
Write the equation of the graph in the form of
�
=
�
�
+
�
y=mx+b that matches.
Line J
�
=
y=
�
+
x +
Line K
�
=
y=
�
+
x+
Line L
�
=
y=
�
+
x+
To determine the equations that match the given graphs, we need to find the slope (m) and y-intercept (b) for each line.
For Line J:
From the given equation, y = -2x + 5, we can see that the slope (m) is -2 and the y-intercept (b) is 5. Therefore, the equation in slope-intercept form is y = -2x + 5.
For Line K:
From the given equation, y = -2x + 3, we can see that the slope (m) is -2 and the y-intercept (b) is 3. Therefore, the equation in slope-intercept form is y = -2x + 3.
For Line L:
Since the equation is not provided for Line L, we cannot determine the exact equation without more information.
Write the equation of the graph in the form of
�
=
�
�
+
�
y=mx+b that matches.
line L
Since the equation of Line L is not provided, we cannot determine the equation in slope-intercept form (y = mx + b) without more information.
Transform
�
−
8
=
2
(
4
�
+
3
)
y−8=2(4x+3) into slope-intercept form (
�
=
�
�
+
�
y=mx+b ).
�
=
y=
�
+
x+
To transform the equation y - 8 = 2(4x + 3) into slope-intercept form (y = mx + b):
Step 1: Distribute the 2 to the terms inside the parentheses:
y - 8 = 8x + 6
Step 2: Move the constant term (-8) to the other side of the equation:
y = 8x + 6 - 8
Simplifying the right side:
y = 8x - 2
So, the equation y - 8 = 2(4x + 3) in slope-intercept form is y = 8x - 2.
Graph the equation y=-3x+5
To graph the equation y = -3x + 5, we can use the slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
In this case, the slope is -3, and the y-intercept is 5.
To plot the graph, follow these steps:
1. Plot the y-intercept: Start at the y-intercept of 5 on the y-axis.
2. Use the slope: From the y-intercept, use the slope -3 to find another point on the line.
To do this, use the slope to determine the change in y and the change in x.
For example, if we move 1 unit to the right along the x-axis, we will move 3 units down along the y-axis.
So, starting from (0, 5), move to the right 1 unit and down 3 units to plot the second point.
3. Connect the two points with a straight line.
The graph of the equation y = -3x + 5 should be a straight line with a slope of -3 and passing through the point (0, 5).
You need to save up to buy a video game system that costs a total of $440 with tax. You currently have $60 in your piggy bank. Your parents will give you $20 per week for allowance and you can work 5 hours per week and make $15 per hour. How many weeks will it take to save up enough money to buy the video game system? Set up an equation and show all steps as you solve the problem.
Let's break down the information given:
Cost of the video game system with tax = $440
Amount currently in the piggy bank = $60
Weekly allowance from parents = $20
Money earned from working 5 hours per week at $15 per hour.
To find out how many weeks it will take to save up enough money, we can set up an equation and solve it step by step.
Let's say the number of weeks needed to save up enough money is represented by "w".
The equation can be set up as:
60 + 20w + 15(5w) = 440
Explanation of each term:
60 represents the amount currently in the piggy bank.
20w represents the total amount received from weekly allowances over w weeks.
15(5w) represents the money earned from working 5 hours per week at $15 per hour over w weeks.
Now, let's solve the equation:
60 + 20w + 15(5w) = 440
Multiply 15 by 5w:
60 + 20w + 75w = 440
Combine like terms:
95w + 60 = 440
Subtract 60 from both sides:
95w = 380
Divide both sides by 95:
w = 4
Therefore, it will take 4 weeks to save up enough money to buy the video game system.
a) Describe how you know that a graph is proportional.
b) Describe how you know an equation is proportional. Give an example of an equation that is proportional and a graph that is not proportional.
c) Describe how you know that a table is proportional.
a) If a graph is proportional, it means that as one variable increases or decreases, the other variable also increases or decreases in a consistent and constant manner. In other words, the graph will form a straight line that passes through the origin (0, 0). This indicates that the two variables are directly proportional to each other.
b) An equation is proportional if it can be expressed in the form y = kx, where y is the dependent variable, x is the independent variable, and k is a constant. The constant k represents the ratio of change between the two variables.
Example of a proportional equation: y = 3x
In this equation, y is directly proportional to x with a constant ratio of 3.
Example of a non-proportional graph: y = x^2
If we graph this equation, we will see a parabola instead of a straight line passing through the origin. This indicates that the graph is not proportional.
c) In a table, if the ratios of the values in one column to the values in another column are the same consistently, then the table is proportional. This means that for each row in the table, the values in the second column will be a constant multiple of the corresponding values in the first column.
For example, consider the following table:
x | y
------------
1 | 2
2 | 4
3 | 6
In this table, we can see that the ratio of y to x is always 2. This indicates that the table is proportional. The values in the second column (y) are twice the corresponding values in the first column (x).
Graph and then find the solution for y=1/2x+2 and y=-1/4x+8 .
The solution to
�
=
1
2
�
+
2
y=
2
1
x+2 and
�
=
−
1
4
�
+
8
y=−
4
1
x+8 is
Let's graph the equations y = 1/2x + 2 and y = -1/4x + 8.
First, let's graph y = 1/2x + 2:
To graph this equation, we need to plot a few points and connect them with a straight line.
For example:
When x = 0, y = 2. So we have the point (0, 2).
When x = 4, y = 4. So we have the point (4, 4).
When x = -4, y = 0. So we have the point (-4, 0).
We can plot these points and draw a straight line through them:
Now, let's graph y = -1/4x + 8:
Again, we'll plot a few points and connect them with a line.
For example:
When x = 0, y = 8. So we have the point (0, 8).
When x = 4, y = 7. So we have the point (4, 7).
When x = -4, y = 9. So we have the point (-4, 9).
We can plot these points and draw a straight line through them:
Now, to find the solution to the system of equations, we need to find the point at which the two lines intersect, if any.
By observing the graph, we can see that the lines intersect at the point (4, 4).
Therefore, the solution to the system of equations is x = 4 and y = 4.
Graph and then find the solution for y=1/2x+2 and y=-1/4x+8 .
The solution to y=1/2x+2 and y=-1/4+8
(__,__)
To find the solution to the system of equations y = (1/2)x + 2 and y = (-1/4)x + 8, we can set the two equations equal to each other and solve for x.
(1/2)x + 2 = (-1/4)x + 8
To eliminate the fractions, we can multiply both sides of the equation by 4:
4[(1/2)x + 2] = 4[(-1/4)x + 8]
2x + 8 = -x + 32
Combine like terms:
2x + x = 32 - 8
3x = 24
Divide both sides by 3:
x = 24 / 3
x = 8
Now, substitute the value of x back into either of the original equations to find the corresponding y-value. Let's use the first equation:
y = (1/2)(8) + 2
y = 4 + 2
y = 6
Therefore, the solution to the system of equations y = (1/2)x + 2 and y = (-1/4)x + 8 is (8, 6).
Graph and then find the solution for y=4/5x-2 and y=-x+7 and .
The solution to y=4/5x-4 and y=-x+7 is
(__,__)
To find the solution to the system of equations y = (4/5)x - 4 and y = -x + 7, we can set the two equations equal to each other and solve for x:
(4/5)x - 4 = -x + 7
To eliminate the fractions, we can multiply both sides of the equation by 5:
5[(4/5)x - 4] = 5[-x + 7]
4x - 20 = -5x + 35
Combine like terms:
4x + 5x = 35 + 20
9x = 55
Divide both sides by 9:
x = 55 / 9
Now, substitute the value of x back into either of the original equations to find the corresponding y-value. Let's use the first equation:
y = (4/5)(55/9) - 4
To simplify, we can multiply the fractions and then subtract 4:
y = (220/45) - 4
To make the fractions have the same denominator, we can multiply 4 by 45/45:
y = (220/45) - (180/45)
y = (220 - 180) / 45
y = 40 / 45
Simplifying the fraction, we can divide both the numerator and denominator by 5:
y = 8 / 9
Therefore, the solution to the system of equations y = (4/5)x - 4 and y = -x + 7 is (55/9, 8/9).