Solved on Nov 10, 2023

Clothing business finds linear relationship between number of shirts $n$ and price $p$ per shirt. Given: 5000 shirts sold at $\$71$ , 6000 shirts sold at $\$65$ . Find linear equation $p=mn+b$ for price $p$ of $n$ shirts.

STEP 1

Assumptions1. The relationship between the number of shirts, $n$ , and the price, $p$ , is linear. . The data points given are (5000,71) and (6000,65), where the first value in the pair is the number of shirts and the second value is the price per shirt.
3. The equation of the line is in the form $p = mn + b$ where $m$ is the slope and $b$ is the y-intercept.

STEP 2

To find the equation of the line, we first need to find the slope, $m$ . The formula for the slope is $m = \frac{y2 - y1}{x2 - x1}$ where $(x1, y1)$ and $(x2, y2)$ are two points on the line.

STEP 3

Substitute the given points into the slope formula.
$m = \frac{65 -71}{6000 -5000}$

STEP 4

Perform the subtraction in the numerator and the denominator to find the slope.
$m = \frac{-6}{1000}$

STEP 5

implify the fraction to find the slope.
$m = -0.006$

STEP 6

Now that we have the slope, we can find the y-intercept, $b$ , by rearranging the equation of the line and substituting one of the points and the slope into the equation.
$b = y - mx$

STEP 7

Substitute the point (5000,71) and the slope into the equation to find the y-intercept.
$b =71 - (-0.006) \times5000$

STEP 8

Perform the multiplication and addition to find the y-intercept.
$b =71 +30 =101$

STEP 9

Now that we have the slope and the y-intercept, we can write the equation of the line.
$p = -.006n +101$ So, the price they can charge for $n$ shirts is given by the equation $p = -.006n +101$ .

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Clothing business finds linear relationship between number of shirts $n$ and price $p$ per shirt. Given: 5000 shirts sold at $\$71$ , 6000 shirts sold at $\$65$ . Find linear equation $p=mn+b$ for price $p$ of $n$ shirts.

STEP 1

STEP 2

To find the equation of the line, we first need to find the slope, $m$ . The formula for the slope is $m = \frac{y2 - y1}{x2 - x1}$ where $(x1, y1)$ and $(x2, y2)$ are two points on the line.

STEP 3

Substitute the given points into the slope formula.
$m = \frac{65 -71}{6000 -5000}$

STEP 4

Perform the subtraction in the numerator and the denominator to find the slope.
$m = \frac{-6}{1000}$

STEP 5

implify the fraction to find the slope.
$m = -0.006$

STEP 6

Now that we have the slope, we can find the y-intercept, $b$ , by rearranging the equation of the line and substituting one of the points and the slope into the equation.
$b = y - mx$

STEP 7

Substitute the point (5000,71) and the slope into the equation to find the y-intercept.
$b =71 - (-0.006) \times5000$

STEP 8

Perform the multiplication and addition to find the y-intercept.
$b =71 +30 =101$

STEP 9

Now that we have the slope and the y-intercept, we can write the equation of the line.
$p = -.006n +101$ So, the price they can charge for $n$ shirts is given by the equation $p = -.006n +101$ .

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Clothing business finds linear relationship between number of shirts nnn and price ppp per shirt. Given: 5000 shirts sold at $71\$71$71, 6000 shirts sold at $65\$65$65. Find linear equation p=mn+bp=mn+bp=mn+b for price ppp of nnn shirts.

STEP 1

STEP 2

To find the equation of the line, we first need to find the slope, mmm. The formula for the slope ism=y2−y1x2−x1m = \frac{y2 - y1}{x2 - x1}m=x2−x1y2−y1​where (x1,y1)(x1, y1)(x1,y1) and (x2,y2)(x2, y2)(x2,y2) are two points on the line.

STEP 3

Substitute the given points into the slope formula.m=65−716000−5000m = \frac{65 -71}{6000 -5000}m=6000−500065−71​

STEP 4

Perform the subtraction in the numerator and the denominator to find the slope.m=−61000m = \frac{-6}{1000}m=1000−6​

STEP 5

implify the fraction to find the slope.m=−0.006m = -0.006m=−0.006

STEP 6

Now that we have the slope, we can find the y-intercept, bbb, by rearranging the equation of the line and substituting one of the points and the slope into the equation.b=y−mxb = y - mxb=y−mx

STEP 7

Substitute the point (5000,71) and the slope into the equation to find the y-intercept.b=71−(−0.006)×5000b =71 - (-0.006) \times5000b=71−(−0.006)×5000

STEP 8

Perform the multiplication and addition to find the y-intercept.b=71+30=101b =71 +30 =101b=71+30=101

STEP 9

Now that we have the slope and the y-intercept, we can write the equation of the line.p=−.006n+101p = -.006n +101p=−.006n+101So, the price they can charge for nnn shirts is given by the equation p=−.006n+101p = -.006n +101p=−.006n+101.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Clothing business finds linear relationship between number of shirts $n$ and price $p$ per shirt. Given: 5000 shirts sold at $\$71$ , 6000 shirts sold at $\$65$ . Find linear equation $p=mn+b$ for price $p$ of $n$ shirts.

To find the equation of the line, we first need to find the slope, $m$ . The formula for the slope is $m = \frac{y2 - y1}{x2 - x1}$ where $(x1, y1)$ and $(x2, y2)$ are two points on the line.

Substitute the given points into the slope formula.
$m = \frac{65 -71}{6000 -5000}$

Perform the subtraction in the numerator and the denominator to find the slope.
$m = \frac{-6}{1000}$

implify the fraction to find the slope.
$m = -0.006$

Now that we have the slope, we can find the y-intercept, $b$ , by rearranging the equation of the line and substituting one of the points and the slope into the equation.
$b = y - mx$

Substitute the point (5000,71) and the slope into the equation to find the y-intercept.
$b =71 - (-0.006) \times5000$

Perform the multiplication and addition to find the y-intercept.
$b =71 +30 =101$

Now that we have the slope and the y-intercept, we can write the equation of the line.
$p = -.006n +101$ So, the price they can charge for $n$ shirts is given by the equation $p = -.006n +101$ .