CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line...
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Transcript of CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line...
![Page 1: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/1.jpg)
CONFIDENTIAL 1
Algebra1Algebra1
Direct VariationDirect Variation
![Page 2: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/2.jpg)
CONFIDENTIAL 2
Warm UpWarm UpFind the slope of the line described by
each equation.
1) 4x + y = -9
2) 6x - 3y = -9
3) 5x = 10y - 5
![Page 3: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/3.jpg)
CONFIDENTIAL 3
Direct Variation
A direct variation is a special type of linear relationship that can be written in the form y = kx, where k is a nonzero constant called the constant of variation .
A recipe for chili calls for 2 pounds of beef to make 4 servings. In other words, a chef needs 2 pounds for every 4 servings.
The equation y = 2x describes this relationship. In this relationship, the number of servings varies directly with the
number of pounds of beef.
Beef (lb) x 2 3 4 5
Servings y 4 6 8 10
![Page 4: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/4.jpg)
CONFIDENTIAL 4
1) y = 4x
Identifying Direct Variations from Equations
Tell whether each equation represents a direct variation. If so, identify the constant of variation.
This equation represents a direct variation because it is in the form y = kx. The constant of variation is 4.
Next Slide
![Page 5: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/5.jpg)
CONFIDENTIAL 5
This equation represents a direct variation because it can be written in the form y = kx. The constant of variation is 3.
5
Since y is multiplied by 5, divide both sides by 5.
5y = 3x 5 5
y = 3x 5
Solve the equation for y.
2) -3x + 5y = 0
-3x + 5y = 0
+3x +3x
5y = 3x
Since -3x is added to y, add 3x to both sides.
Next Slide
![Page 6: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/6.jpg)
CONFIDENTIAL 6
Solve the equation for y.
Since 2x is added to y, subtract 2x from both sides.y =-2x + 10
This equation does not represent a direct variation because it cannot be written in the form y = kx.
3) 2x + y = 10
2x + y = 10
-2x -2x
![Page 7: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/7.jpg)
CONFIDENTIAL 7
Now you try!
1a) 3y = 4x + 1
1b) 3x = -4y
1c) y + 3x = 0
Tell whether each equation represents a direct variation. If so, identify the constant of variation.
![Page 8: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/8.jpg)
CONFIDENTIAL 8
What happens if you solve y = kx for k?
y = kxx x
y = kx
So, in a direct variation, the ratio y is equal to the x
constant of variation.
Another way to identify a direct variation is to check
whether y is the same for each ordered pair
(except where x = 0).
x
Divide both sides by x (x ≠ 0)
y = kx
Divide both sides by x (x ≠ 0)
![Page 9: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/9.jpg)
CONFIDENTIAL 9
Identifying Direct Variations from Ordered Pairs
Tell whether each relationship is a direct variation. Explain.
x 1 3 5
y 6 18 30
1)
Method 1: Write an equation.
Each y-value is 6 times the corresponding x-value.y = 6x
This is a direct variation because it can be written as y = kx, where k = 6.
6 = 61
This is a direct variation because y is the same for xeach ordered pair.
Method 2: Find y for each ordered pair.x
18 = 6 3
30 = 6 5
![Page 10: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/10.jpg)
CONFIDENTIAL 10
x 2 4 8
y -2 0 4
2)
Method 1: Write an equation.
Each y-value is 4 less than the corresponding x-value.y = x - 4
This is not a direct variation because it cannot be written as y = kx.
-2 = -1 2
This is not a direct variation because y is not the same xfor each ordered pair.
Method 2: Find y for each ordered pair.x
0 = 04
4 = 18 2
![Page 11: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/11.jpg)
CONFIDENTIAL 11
Now you try!
2c)
Tell whether each relationship is a direct variation. Explain.
2a) 2b)
![Page 12: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/12.jpg)
CONFIDENTIAL 12
If you know one ordered pair that satisfies a direct variation, you can write the
equation. You can also find other ordered pairs that satisfy the direct variation.
![Page 13: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/13.jpg)
CONFIDENTIAL 13
Writing and Solving Direct Variation Equations
The value of y varies directly with x, and y = 6 when x = 12.Find y when x = 27.
Method 1: Find the value of k and then write the equation.
Write the equation for a direct variation.y = kx
The equation is y = 1 x . When x = 27, y = 1 (27) = 13.5. 2 2
Substitute 6 for y and 12 for x. Solve for k.6 = k(12)
Since k is multiplied by 12, divide both sides by 12.6 = k(12)
![Page 14: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/14.jpg)
CONFIDENTIAL 14
Method 2: Use a proportion.
In a direct variation, y xis the same for all values of x and y.
6 = y12 27
Use cross products.12y = 162
Since y is multiplied by 12, divide both sides by 12.y = 13.5
![Page 15: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/15.jpg)
CONFIDENTIAL 15
Now you try!
3) The value of y varies directly with x, and y = 4.5 when x = 0.5. Find y when x = 10.
![Page 16: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/16.jpg)
CONFIDENTIAL 16
Graphing Direct Variations
1) The three-toed sloth is an extremely slow animal. On the ground, it travels at a speed of about 6 feet per
minute. Write a direct variation equation for the distance y a sloth will travel in x minutes. Then graph.
Step1: Write a direct variation equation.
distance = 6 feet per minute
Step2: Choose values of x and generate ordered pairs.
X y = 6x (x, y)
0 y = 6 (0) = 0 (0, 0)
1 y = 6 (1) = 6 (1, 6)
2 y = 6 (2) = 12 (2, 12)
![Page 17: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/17.jpg)
CONFIDENTIAL 17
Step3: Graph the points and connect.
Look at the graph. It passes through (0, 0) and has a slope of 6. The graph of any direct variation y = kx
• is a line through (0, 0) • has a slope of k.
![Page 18: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/18.jpg)
CONFIDENTIAL 18
Now you try!
4) The perimeter y of a square varies directly with its side length x. Write a direct variation equation
for this relationship. Then graph.
![Page 19: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/19.jpg)
CONFIDENTIAL 19
Assessment
1) y = 4x + 9
Tell whether each equation represents a direct variation. If so, identify the constant of variation.
2) 2y = -8x
3) x + y = 0
![Page 20: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/20.jpg)
CONFIDENTIAL 20
Tell whether each relationship is a direct variation. Explain.
5)
4)
![Page 21: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/21.jpg)
CONFIDENTIAL 21
6) The value of y varies directly with x, and y = -3 when x = 1. Find y when x = -6.
7) The value of y varies directly with x, and y = 6 when x = 18. Find y when x = 12.
![Page 22: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/22.jpg)
CONFIDENTIAL 22
8) Cameron earns $5 per hour at her after-school job. The total amount of her paycheck varies directly with the amount of time she works. Write a direct variation equation for the amount of money y that she earns for working x hours. Then graph.
![Page 23: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/23.jpg)
CONFIDENTIAL 23
9) (2, 10)
10) (-3, 9)
Each ordered pair is a solution of a direct variation. Write the equation of direct variation. Then graph
your equation and show that the slope of the line is equal to the constant of variation.
![Page 24: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/24.jpg)
CONFIDENTIAL 24
Direct Variation
A direct variation is a special type of linear relationship that can be written in the form y = kx, where k is a nonzero constant called the constant of variation .
A recipe for chili calls for 2 pounds of beef to make 4 servings. In other words, a chef needs 2 pounds for every 4 servings.
The equation y = 2x describes this relationship. In this relationship, the number of servings varies directly with the
number of pounds of beef.
Beef (lb) x 2 3 4 5
Servings y 4 6 8 10
Let’s review
![Page 25: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/25.jpg)
CONFIDENTIAL 25
This equation represents a direct variation because it can be written in the form y = kx. The constant of variation is 3.
5
Since y is multiplied by 5, divide both sides by 5.
5y = 3x 5 5
y = 3x 5
Solve the equation for y.
2) -3x + 5y = 0
-3x + 5y = 0
+3x +3x
5y = 3x
Since -3x is added to y, add 3x to both sides.
Next Slide
Identifying Direct Variations from Equations
Tell whether each equation represents a direct variation. If so, identify the constant of variation.
![Page 26: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/26.jpg)
CONFIDENTIAL 26
Solve the equation for y.
Since 2x is added to y, subtract 2x from both sides.y =-2x + 10
This equation does not represent a direct variation because it cannot be written in the form y = kx.
3) 2x + y = 10
2x + y = 10
-2x -2x
![Page 27: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/27.jpg)
CONFIDENTIAL 27
What happens if you solve y = kx for k?
y = kxx x
y = kx
So, in a direct variation, the ratio y is equal to the x
constant of variation.
Another way to identify a direct variation is to check
whether y is the same for each ordered pair
(except where x = 0).
x
Divide both sides by x (x ≠ 0)
y = kx
Divide both sides by x (x ≠ 0)
![Page 28: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/28.jpg)
CONFIDENTIAL 28
Identifying Direct Variations from Ordered Pairs
Tell whether each relationship is a direct variation. Explain.
x 1 3 5
y 6 18 30
1)
Method 1: Write an equation.
Each y-value is 6 times the corresponding x-value.y = 6x
This is a direct variation because it can be written as y = kx, where k = 6.
6 = 61
This is a direct variation because y is the same for xeach ordered pair.
Method 2: Find y for each ordered pair.x
18 = 6 3
30 = 6 5
![Page 29: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/29.jpg)
CONFIDENTIAL 29
Writing and Solving Direct Variation Equations
The value of y varies directly with x, and y = 6 when x = 12.Find y when x = 27.
Method 1: Find the value of k and then write the equation.
Write the equation for a direct variation.y = kx
The equation is y = 1 x . When x = 27, y = 1 (27) = 13.5. 2 2
Substitute 6 for y and 12 for x. Solve for k.6 = k(12)
Since k is multiplied by 12, divide both sides by 12.6 = k(12)
![Page 30: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/30.jpg)
CONFIDENTIAL 30
Method 2: Use a proportion.
In a direct variation, y xis the same for all values of x and y.
6 = y12 27
Use cross products.12y = 162
Since y is multiplied by 12, divide both sides by 12.y = 13.5
![Page 31: CONFIDENTIAL 1 Algebra1 Direct Variation. CONFIDENTIAL 2 Warm Up Find the slope of the line described by each equation. 1) 4x + y = -9 2) 6x - 3y = -9.](https://reader035.fdocuments.in/reader035/viewer/2022062500/5697bffa1a28abf838cc01a9/html5/thumbnails/31.jpg)
CONFIDENTIAL 31
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