QuestionFind the slope $m$ and y-intercept $b$ of the linear function $f(x)=m x+b$ given $f(-5)=16$ and $f(3)=-8$ .

Studdy Solution

STEP 1

Assumptions1. The function is linear, represented as $f(x) = mx + b$ . We know two points on the line $(-5,16)$ and $(3, -8)$ 3. We need to find the slope $m$ and the y-intercept $b$

STEP 2

We can use the two points given to form two equations. The first equation is formed by substituting $x = -5$ and $f(-5) =16$ into the equation $f(x) = mx + b$ .
$16 = -5m + b$

STEP 3

The second equation is formed by substituting $x =3$ and $f(3) = -8$ into the equation $f(x) = mx + b$ .
$-8 =3m + b$

STEP 4

We now have a system of two equations\begin{align} 16 &= -m + b\\-8 &=3m + b\end{align}

STEP 5

We can solve for $m$ and $b$ by subtracting the second equation from the first.
$16 - (-8) = -5m -3m + b - b$

STEP 6

implify the left side of the equation.
$24 = -8m$

STEP 7

olve for $m$ by dividing both sides of the equation by $-$ .
$m = \frac{24}{-} = -3$

STEP 8

Now that we have $m$ , we can substitute it into one of the original equations to solve for $b$ . Let's use the first equation $16 = -5m + b$ .
$16 = -5(-3) + b$

STEP 9

implify the right side of the equation.
$16 =15 + b$

STEP 10

olve for $b$ by subtracting $15$ from both sides of the equation.
$b =16 -15 =$ So, $m = -3$ and $b =$ .

Was this helpful?

QuestionFind the slope $m$ and y-intercept $b$ of the linear function $f(x)=m x+b$ given $f(-5)=16$ and $f(3)=-8$ .

Studdy Solution

STEP 1

Assumptions1. The function is linear, represented as $f(x) = mx + b$ . We know two points on the line $(-5,16)$ and $(3, -8)$ 3. We need to find the slope $m$ and the y-intercept $b$

STEP 2

We can use the two points given to form two equations. The first equation is formed by substituting $x = -5$ and $f(-5) =16$ into the equation $f(x) = mx + b$ .
$16 = -5m + b$

STEP 3

The second equation is formed by substituting $x =3$ and $f(3) = -8$ into the equation $f(x) = mx + b$ .
$-8 =3m + b$

STEP 4

We now have a system of two equations\begin{align} 16 &= -m + b\\-8 &=3m + b\end{align}

STEP 5

We can solve for $m$ and $b$ by subtracting the second equation from the first.
$16 - (-8) = -5m -3m + b - b$

STEP 6

implify the left side of the equation.
$24 = -8m$

STEP 7

olve for $m$ by dividing both sides of the equation by $-$ .
$m = \frac{24}{-} = -3$

STEP 8

Now that we have $m$ , we can substitute it into one of the original equations to solve for $b$ . Let's use the first equation $16 = -5m + b$ .
$16 = -5(-3) + b$

STEP 9

implify the right side of the equation.
$16 =15 + b$

STEP 10

olve for $b$ by subtracting $15$ from both sides of the equation.
$b =16 -15 =$ So, $m = -3$ and $b =$ .

Get Studdy

Studdy solves anything!

Start learning now

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

QuestionFind the slope mmm and y-intercept bbb of the linear function f(x)=mx+bf(x)=m x+bf(x)=mx+b given f(−5)=16f(-5)=16f(−5)=16 and f(3)=−8f(3)=-8f(3)=−8.

Studdy Solution

STEP 1

Assumptions1. The function is linear, represented as f(x)=mx+bf(x) = mx + bf(x)=mx+b . We know two points on the line (−5,16)(-5,16)(−5,16) and (3,−8)(3, -8)(3,−8)3. We need to find the slope mmm and the y-intercept bbb

STEP 2

We can use the two points given to form two equations. The first equation is formed by substituting x=−5x = -5x=−5 and f(−5)=16f(-5) =16f(−5)=16 into the equation f(x)=mx+bf(x) = mx + bf(x)=mx+b.16=−5m+b16 = -5m + b16=−5m+b

STEP 3

The second equation is formed by substituting x=3x =3x=3 and f(3)=−8f(3) = -8f(3)=−8 into the equation f(x)=mx+bf(x) = mx + bf(x)=mx+b.−8=3m+b-8 =3m + b−8=3m+b

STEP 4

We now have a system of two equations\begin{align*} 16 &= -m + b\\-8 &=3m + b\end{align*}

STEP 5

We can solve for mmm and bbb by subtracting the second equation from the first.16−(−8)=−5m−3m+b−b16 - (-8) = -5m -3m + b - b16−(−8)=−5m−3m+b−b

STEP 6

implify the left side of the equation.24=−8m24 = -8m24=−8m

STEP 7

olve for mmm by dividing both sides of the equation by −-−.m=24−=−3m = \frac{24}{-} = -3m=−24​=−3

STEP 8

Now that we have mmm, we can substitute it into one of the original equations to solve for bbb. Let's use the first equation 16=−5m+b16 = -5m + b16=−5m+b.16=−5(−3)+b16 = -5(-3) + b16=−5(−3)+b

STEP 9

implify the right side of the equation.16=15+b16 =15 + b16=15+b

STEP 10

olve for bbb by subtracting 151515 from both sides of the equation.b=16−15=b =16 -15 =b=16−15=So, m=−3m = -3m=−3 and b=b =b=.

Get Studdy

Studdy solves anything!

Start learning now

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

QuestionFind the slope $m$ and y-intercept $b$ of the linear function $f(x)=m x+b$ given $f(-5)=16$ and $f(3)=-8$ .

Assumptions1. The function is linear, represented as $f(x) = mx + b$ . We know two points on the line $(-5,16)$ and $(3, -8)$ 3. We need to find the slope $m$ and the y-intercept $b$

We can use the two points given to form two equations. The first equation is formed by substituting $x = -5$ and $f(-5) =16$ into the equation $f(x) = mx + b$ .
$16 = -5m + b$

The second equation is formed by substituting $x =3$ and $f(3) = -8$ into the equation $f(x) = mx + b$ .
$-8 =3m + b$

We now have a system of two equations\begin{align} 16 &= -m + b\\-8 &=3m + b\end{align}

We can solve for $m$ and $b$ by subtracting the second equation from the first.
$16 - (-8) = -5m -3m + b - b$

implify the left side of the equation.
$24 = -8m$

olve for $m$ by dividing both sides of the equation by $-$ .
$m = \frac{24}{-} = -3$

Now that we have $m$ , we can substitute it into one of the original equations to solve for $b$ . Let's use the first equation $16 = -5m + b$ .
$16 = -5(-3) + b$

implify the right side of the equation.
$16 =15 + b$

olve for $b$ by subtracting $15$ from both sides of the equation.
$b =16 -15 =$ So, $m = -3$ and $b =$ .