QuestionBob bought a house for \$42,000. Now it's worth \$67,500 after 8 years.
(a) Find the linear equation $V=m t+b$ for $0 \leq t \leq 15$ . What are $m$ and $b$ ? (b) Estimate when the house will be worth \$72,500. (c) Solve for when it will be worth \$74,000. (d) Find when it will be worth \$80,250.

Studdy Solution

STEP 1

Assumptions1. The initial purchase price of the house is $42,000. . The house was appraised8 years later at$ 67,500.
3. The value of the house increases linearly over time.
4. The linear equation representing the value of the house is $V=mt+b$ , where $V$ is the value of the house, $m$ is the slope of the line (rate of increase in value per year), $t$ is the time in years, and $b$ is the y-intercept (initial value of the house).

STEP 2

First, we need to find the slope $m$ of the line. The slope is the rate of change in the value of the house per year. It can be calculated by the formula $m = \frac{V_{final} - V_{initial}}{t_{final} - t_{initial}}$

STEP 3

Now, plug in the given values for the final value ( $V_{final}$ ), initial value ( $V_{initial}$ ), final time ( $t_{final}$ ), and initial time ( $t_{initial}$ ) to calculate the slope.
$m = \frac{\$67,500 - \$42,000}{8\,years -0\,years}$

STEP 4

Calculate the slope.
$m = \frac{\$67,500 - \$42,000}{8\,years} = \$3,187.50/year$

STEP 5

The y-intercept $b$ is the initial value of the house, which is $42,000. So, the linear equation representing the value of the house is$ $V = \$3,187.50t + \$42,000$ $

STEP 6

To estimate when the house will be worth $72,500, we can graph the equation and trace the line. However, as an AI, I can't graph the equation. Instead, I'll solve the equation algebraically.

STEP 7

Set $V$ to $72,500 and solve for$ t$.
$\$72,500 = \$3,187.50t + \$42,000$

STEP 8

Subtract $42,000 from both sides of the equation.
$\$30,500 = \$3,187.50t$

STEP 9

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$30,500}{\$3,187.50}$

STEP 10

Calculate $t$ .
$t = \frac{\$30,500}{\$3,187.50} =9.57\,years$

STEP 11

To find when the house will be worth $74,000, set$ V $to$ 74,000 and solve for $t$ .
$\$74,000 = \$3,187.50t + \$42,000$

STEP 12

Subtract $42,000 from both sides of the equation.
$\$32,000 = \$,187.50t$

STEP 13

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$32,000}{\$3,187.50}$

STEP 14

Calculate $t$ .
$t = \frac{\$32,000}{\$3,187.50} =10.04\,years$

STEP 15

To find when the house will be worth $80,250, set$ V $to$ 80,250 and solve for $t$ .
$\$80,250 = \$3,187.50t + \$42,000$

STEP 16

Subtract $42,000 from both sides of the equation.
$\$38,250 = \$3,187.50t$

STEP 17

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$38,250}{\$3,187.50}$

STEP 18

Calculate $t$ .
$t = \frac{\$38,250}{\$3,187.50} =12\,years$ So, the house will be worth $72,500 in about.57 years,$ 74,000 in about10.04 years, and $80,250 in about12 years after purchase.

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Studdy Solution

STEP 1

STEP 2

First, we need to find the slope $m$ of the line. The slope is the rate of change in the value of the house per year. It can be calculated by the formula $m = \frac{V_{final} - V_{initial}}{t_{final} - t_{initial}}$

STEP 3

Now, plug in the given values for the final value ( $V_{final}$ ), initial value ( $V_{initial}$ ), final time ( $t_{final}$ ), and initial time ( $t_{initial}$ ) to calculate the slope.
$m = \frac{\$67,500 - \$42,000}{8\,years -0\,years}$

STEP 4

Calculate the slope.
$m = \frac{\$67,500 - \$42,000}{8\,years} = \$3,187.50/year$

STEP 5

The y-intercept $b$ is the initial value of the house, which is $42,000. So, the linear equation representing the value of the house is$ $V = \$3,187.50t + \$42,000$ $

STEP 6

To estimate when the house will be worth $72,500, we can graph the equation and trace the line. However, as an AI, I can't graph the equation. Instead, I'll solve the equation algebraically.

STEP 7

Set $V$ to $72,500 and solve for$ t$.
$\$72,500 = \$3,187.50t + \$42,000$

STEP 8

Subtract $42,000 from both sides of the equation.
$\$30,500 = \$3,187.50t$

STEP 9

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$30,500}{\$3,187.50}$

STEP 10

Calculate $t$ .
$t = \frac{\$30,500}{\$3,187.50} =9.57\,years$

STEP 11

To find when the house will be worth $74,000, set$ V $to$ 74,000 and solve for $t$ .
$\$74,000 = \$3,187.50t + \$42,000$

STEP 12

Subtract $42,000 from both sides of the equation.
$\$32,000 = \$,187.50t$

STEP 13

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$32,000}{\$3,187.50}$

STEP 14

Calculate $t$ .
$t = \frac{\$32,000}{\$3,187.50} =10.04\,years$

STEP 15

To find when the house will be worth $80,250, set$ V $to$ 80,250 and solve for $t$ .
$\$80,250 = \$3,187.50t + \$42,000$

STEP 16

Subtract $42,000 from both sides of the equation.
$\$38,250 = \$3,187.50t$

STEP 17

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$38,250}{\$3,187.50}$

STEP 18

Calculate $t$ .
$t = \frac{\$38,250}{\$3,187.50} =12\,years$ So, the house will be worth $72,500 in about.57 years,$ 74,000 in about10.04 years, and $80,250 in about12 years after purchase.

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Studdy Solution

STEP 1

STEP 2

First, we need to find the slope mmm of the line. The slope is the rate of change in the value of the house per year. It can be calculated by the formulam=Vfinal−Vinitialtfinal−tinitialm = \frac{V_{final} - V_{initial}}{t_{final} - t_{initial}}m=tfinal​−tinitial​Vfinal​−Vinitial​​

STEP 3

STEP 4

Calculate the slope.m=$67,500−$42,0008 years=$3,187.50/yearm = \frac{\$67,500 - \$42,000}{8\,years} = \$3,187.50/yearm=8years$67,500−$42,000​=$3,187.50/year

STEP 5

STEP 6

To estimate when the house will be worth $72,500, we can graph the equation and trace the line. However, as an AI, I can't graph the equation. Instead, I'll solve the equation algebraically.

STEP 7

Set VVV to 72,500andsolvefor72,500 and solve for 72,500andsolvefort$.$72,500=$3,187.50t+$42,000\$72,500 = \$3,187.50t + \$42,000$72,500=$3,187.50t+$42,000

STEP 8

Subtract $42,000 from both sides of the equation.$30,500=$3,187.50t\$30,500 = \$3,187.50t$30,500=$3,187.50t

STEP 9

Divide both sides of the equation by 3,187.50tosolvefor3,187.50 to solve for 3,187.50tosolvefort$.t=$30,500$3,187.50t = \frac{\$30,500}{\$3,187.50}t=$3,187.50$30,500​

STEP 10

Calculate ttt.t=$30,500$3,187.50=9.57 yearst = \frac{\$30,500}{\$3,187.50} =9.57\,yearst=$3,187.50$30,500​=9.57years

STEP 11

To find when the house will be worth 74,000,set74,000, set 74,000,setVto to to74,000 and solve for ttt.$74,000=$3,187.50t+$42,000\$74,000 = \$3,187.50t + \$42,000$74,000=$3,187.50t+$42,000

STEP 12

Subtract $42,000 from both sides of the equation.$32,000=$,187.50t\$32,000 = \$,187.50t$32,000=$,187.50t

STEP 13

Divide both sides of the equation by 3,187.50tosolvefor3,187.50 to solve for 3,187.50tosolvefort$.t=$32,000$3,187.50t = \frac{\$32,000}{\$3,187.50}t=$3,187.50$32,000​

STEP 14

Calculate ttt.t=$32,000$3,187.50=10.04 yearst = \frac{\$32,000}{\$3,187.50} =10.04\,yearst=$3,187.50$32,000​=10.04years

STEP 15

To find when the house will be worth 80,250,set80,250, set 80,250,setVto to to80,250 and solve for ttt.$80,250=$3,187.50t+$42,000\$80,250 = \$3,187.50t + \$42,000$80,250=$3,187.50t+$42,000

STEP 16

Subtract $42,000 from both sides of the equation.$38,250=$3,187.50t\$38,250 = \$3,187.50t$38,250=$3,187.50t

STEP 17

Divide both sides of the equation by 3,187.50tosolvefor3,187.50 to solve for 3,187.50tosolvefort$.t=$38,250$3,187.50t = \frac{\$38,250}{\$3,187.50}t=$3,187.50$38,250​

STEP 18

Calculate ttt.t=$38,250$3,187.50=12 yearst = \frac{\$38,250}{\$3,187.50} =12\,yearst=$3,187.50$38,250​=12yearsSo, the house will be worth 72,500inabout.57years,72,500 in about.57 years, 72,500inabout.57years,74,000 in about10.04 years, and $80,250 in about12 years after purchase.

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First, we need to find the slope $m$ of the line. The slope is the rate of change in the value of the house per year. It can be calculated by the formula $m = \frac{V_{final} - V_{initial}}{t_{final} - t_{initial}}$

Calculate the slope.
$m = \frac{\$67,500 - \$42,000}{8\,years} = \$3,187.50/year$

Set $V$ to $72,500 and solve for$ t$.
$\$72,500 = \$3,187.50t + \$42,000$

Subtract $42,000 from both sides of the equation.
$\$30,500 = \$3,187.50t$

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$30,500}{\$3,187.50}$

Calculate $t$ .
$t = \frac{\$30,500}{\$3,187.50} =9.57\,years$

To find when the house will be worth $74,000, set$ V $to$ 74,000 and solve for $t$ .
$\$74,000 = \$3,187.50t + \$42,000$

Subtract $42,000 from both sides of the equation.
$\$32,000 = \$,187.50t$

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$32,000}{\$3,187.50}$

Calculate $t$ .
$t = \frac{\$32,000}{\$3,187.50} =10.04\,years$

To find when the house will be worth $80,250, set$ V $to$ 80,250 and solve for $t$ .
$\$80,250 = \$3,187.50t + \$42,000$

Subtract $42,000 from both sides of the equation.
$\$38,250 = \$3,187.50t$

Divide both sides of the equation by $3,187.50 to solve for$ t$.
$t = \frac{\$38,250}{\$3,187.50}$

Calculate $t$ .
$t = \frac{\$38,250}{\$3,187.50} =12\,years$ So, the house will be worth $72,500 in about.57 years,$ 74,000 in about10.04 years, and $80,250 in about12 years after purchase.