Questions: Find all values of x satisfying the given conditions. y1=5(2 x-3)-9, y2=8(x-4)+22, y1=y2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The value(s) of x for which y1=y2 is/are . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. y1=y2 for all values of x. C. There are no values of x for which y1=y2.

$Find all values of x satisfying the given conditions. y1=5(2 x-3)-9, y2=8(x-4)+22, y1=y2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The value(s) of x for which y1=y2 is/are . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. y1=y2 for all values of x. C. There are no values of x for which y1=y2.$

Transcript text: Find all values of x satisfying the given conditions. \[ y_{1}=5(2 x-3)-9, y_{2}=8(x-4)+22, y_{1}=y_{2} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The value(s) of x for which $\mathrm{y}_{1}=\mathrm{y}_{2}$ is/are $\}$. $\square$ (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. $y_{1}=y_{2}$ for all values of $x$. C. There are no values of x for which $\mathrm{y}_{1}=\mathrm{y}_{2}$.

Solution

Solution Steps

Step 1: Expand $ y_1 $ and $ y_2 $

Expand the expressions for $ y_1 $ and $ y_2 $: \[ y_1 = 5(2x - 3) - 9 = 10x - 15 - 9 = 10x - 24 \] \[ y_2 = 8(x - 4) + 22 = 8x - 32 + 22 = 8x - 10 \]

Step 2: Set $ y_1 = y_2 $ and solve for $ x $

Set the expanded expressions equal to each other: \[ 10x - 24 = 8x - 10 \] Subtract $ 8x $ from both sides: \[ 2x - 24 = -10 \] Add $ 24 $ to both sides: \[ 2x = 14 \] Divide both sides by $ 2 $: \[ x = 7 \]

Step 3: Verify the solution

Substitute $ x = 7 $ back into the original expressions for $ y_1 $ and $ y_2 $ to verify: \[ y_1 = 5(2(7) - 3) - 9 = 5(14 - 3) - 9 = 5(11) - 9 = 55 - 9 = 46 \] \[ y_2 = 8(7 - 4) + 22 = 8(3) + 22 = 24 + 22 = 46 \] Since $ y_1 = y_2 $ when $ x = 7 $, the solution is correct.

Step 4: Select the correct choice

The value of $ x $ for which $ y_1 = y_2 $ is $ 7 $. Therefore, the correct choice is: \[ \text{A. The value(s) of } x \text{ for which } y_1 = y_2 \text{ is/are } \boxed{7}. \]

The remaining parts of the question are left unaddressed as per the guidelines.