Solved on Jan 20, 2024

Find the value of $A$ if $g(t) = \frac{2t - 9}{t - A}$ and $g(3) = 3$ . $A = \square$

STEP 1

Assumptions
1. The function is given by $g(t)=\frac{2t-9}{t-A}$ .
2. The value of the function at $t=3$ is $g(3)=3$ .
3. We need to find the value of $A$ .

STEP 2

Substitute $t=3$ into the function $g(t)$ to find the equation that will allow us to solve for $A$ .
$g(3) = \frac{2(3)-9}{3-A}$

STEP 3

Now, plug in the given value for $g(3)$ into the equation.
$3 = \frac{2(3)-9}{3-A}$

STEP 4

Simplify the numerator of the fraction on the right side of the equation.
$3 = \frac{6-9}{3-A}$

STEP 5

Further simplify the numerator by performing the subtraction.
$3 = \frac{-3}{3-A}$

STEP 6

Since $g(3)=3$ , we can set up the equation as follows:
$3 = \frac{-3}{3-A}$

STEP 7

To solve for $A$ , we can multiply both sides of the equation by the denominator $(3-A)$ to get rid of the fraction.
$3(3-A) = -3$

STEP 8

Distribute the $3$ on the left side of the equation.
$9 - 3A = -3$

STEP 9

Add $3A$ to both sides of the equation to move the term involving $A$ to the right side.
$9 = -3 + 3A$

STEP 10

Add $3$ to both sides of the equation to isolate the term with $A$ on the right side.
$9 + 3 = 3A$

STEP 11

Combine the constants on the left side of the equation.
$12 = 3A$

STEP 12

Divide both sides of the equation by $3$ to solve for $A$ .
$A = \frac{12}{3}$

STEP 13

Calculate the value of $A$ .
$A = 4$
The value of $A$ is $4$ .

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Find the value of $A$ if $g(t) = \frac{2t - 9}{t - A}$ and $g(3) = 3$ . $A = \square$

STEP 1

Assumptions
1. The function is given by $g(t)=\frac{2t-9}{t-A}$ .
2. The value of the function at $t=3$ is $g(3)=3$ .
3. We need to find the value of $A$ .

STEP 2

Substitute $t=3$ into the function $g(t)$ to find the equation that will allow us to solve for $A$ .
$g(3) = \frac{2(3)-9}{3-A}$

STEP 3

Now, plug in the given value for $g(3)$ into the equation.
$3 = \frac{2(3)-9}{3-A}$

STEP 4

Simplify the numerator of the fraction on the right side of the equation.
$3 = \frac{6-9}{3-A}$

STEP 5

Further simplify the numerator by performing the subtraction.
$3 = \frac{-3}{3-A}$

STEP 6

Since $g(3)=3$ , we can set up the equation as follows:
$3 = \frac{-3}{3-A}$

STEP 7

To solve for $A$ , we can multiply both sides of the equation by the denominator $(3-A)$ to get rid of the fraction.
$3(3-A) = -3$

STEP 8

Distribute the $3$ on the left side of the equation.
$9 - 3A = -3$

STEP 9

Add $3A$ to both sides of the equation to move the term involving $A$ to the right side.
$9 = -3 + 3A$

STEP 10

Add $3$ to both sides of the equation to isolate the term with $A$ on the right side.
$9 + 3 = 3A$

STEP 11

Combine the constants on the left side of the equation.
$12 = 3A$

STEP 12

Divide both sides of the equation by $3$ to solve for $A$ .
$A = \frac{12}{3}$

STEP 13

Calculate the value of $A$ .
$A = 4$
The value of $A$ is $4$ .

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Find the value of AAA if g(t)=2t−9t−Ag(t) = \frac{2t - 9}{t - A}g(t)=t−A2t−9​ and g(3)=3g(3) = 3g(3)=3. A=□A = \squareA=□

STEP 1

Assumptions1. The function is given by g(t)=2t−9t−Ag(t)=\frac{2t-9}{t-A}g(t)=t−A2t−9​.2. The value of the function at t=3t=3t=3 is g(3)=3g(3)=3g(3)=3.3. We need to find the value of AAA.

STEP 2

Substitute t=3t=3t=3 into the function g(t)g(t)g(t) to find the equation that will allow us to solve for AAA.g(3)=2(3)−93−Ag(3) = \frac{2(3)-9}{3-A}g(3)=3−A2(3)−9​

STEP 3

Now, plug in the given value for g(3)g(3)g(3) into the equation.3=2(3)−93−A3 = \frac{2(3)-9}{3-A}3=3−A2(3)−9​

STEP 4

Simplify the numerator of the fraction on the right side of the equation.3=6−93−A3 = \frac{6-9}{3-A}3=3−A6−9​

STEP 5

Further simplify the numerator by performing the subtraction.3=−33−A3 = \frac{-3}{3-A}3=3−A−3​

STEP 6

Since g(3)=3g(3)=3g(3)=3, we can set up the equation as follows:3=−33−A3 = \frac{-3}{3-A}3=3−A−3​

STEP 7

To solve for AAA, we can multiply both sides of the equation by the denominator (3−A)(3-A)(3−A) to get rid of the fraction.3(3−A)=−33(3-A) = -33(3−A)=−3

STEP 8

Distribute the 333 on the left side of the equation.9−3A=−39 - 3A = -39−3A=−3

STEP 9

Add 3A3A3A to both sides of the equation to move the term involving AAA to the right side.9=−3+3A9 = -3 + 3A9=−3+3A

STEP 10

Add 333 to both sides of the equation to isolate the term with AAA on the right side.9+3=3A9 + 3 = 3A9+3=3A

STEP 11

Combine the constants on the left side of the equation.12=3A12 = 3A12=3A

STEP 12

Divide both sides of the equation by 333 to solve for AAA.A=123A = \frac{12}{3}A=312​

STEP 13

Calculate the value of AAA.A=4A = 4A=4The value of AAA is 444.

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Find the value of $A$ if $g(t) = \frac{2t - 9}{t - A}$ and $g(3) = 3$ . $A = \square$

Assumptions
1. The function is given by $g(t)=\frac{2t-9}{t-A}$ .
2. The value of the function at $t=3$ is $g(3)=3$ .
3. We need to find the value of $A$ .

Substitute $t=3$ into the function $g(t)$ to find the equation that will allow us to solve for $A$ .
$g(3) = \frac{2(3)-9}{3-A}$

Now, plug in the given value for $g(3)$ into the equation.
$3 = \frac{2(3)-9}{3-A}$

Simplify the numerator of the fraction on the right side of the equation.
$3 = \frac{6-9}{3-A}$

Further simplify the numerator by performing the subtraction.
$3 = \frac{-3}{3-A}$

Since $g(3)=3$ , we can set up the equation as follows:
$3 = \frac{-3}{3-A}$

To solve for $A$ , we can multiply both sides of the equation by the denominator $(3-A)$ to get rid of the fraction.
$3(3-A) = -3$

Distribute the $3$ on the left side of the equation.
$9 - 3A = -3$

Add $3A$ to both sides of the equation to move the term involving $A$ to the right side.
$9 = -3 + 3A$

Add $3$ to both sides of the equation to isolate the term with $A$ on the right side.
$9 + 3 = 3A$

Combine the constants on the left side of the equation.
$12 = 3A$

Divide both sides of the equation by $3$ to solve for $A$ .
$A = \frac{12}{3}$

Calculate the value of $A$ .
$A = 4$
The value of $A$ is $4$ .