the line containing the points (-3,k) and (4,8) is parallel to the line containing the points (5,3) and (1,-6) what is the value of k ?

To determine the value of k, the first step we need to do is to find the slope of the line containing the points (5,3) and (1,-6) using the formula

[tex]\mathrm{Slope}=m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Here:

[tex]\left(x_1,\:y_1\right)=\left(5,\:3\right)[/tex]

[tex]\left(x_2,\:y_2\right)=\left(1,\:-6\right)[/tex]

now substituting (x₁, y₁) = (5, 3) and (x₂, y₂) = (1, -6) in the formula

[tex]m=\frac{-6-3}{1-5}[/tex]

[tex]m=\frac{-9}{-4}[/tex]

Apply the fraction rule: [tex]\frac{-a}{-b}=\frac{a}{b}[/tex]

[tex]m=\frac{9}{4}[/tex]

Thus, we conclude that the slope of the line containing the points (5,3) and (1,-6) will be:

[tex]m=\frac{9}{4}[/tex]

Importing Tip:

We know that the parallel lines have the same slopes, so the slope of the line containing the points (-3,k) and (4,8) will also be 9/4.

Thus, using the slope formula for the line containing the points (-3,k) and (4,8).

[tex]\mathrm{Slope}=m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Here:

[tex]\left(x_1,\:y_1\right)=\left(-3,\:k\right)[/tex]

[tex]\left(x_2,\:y_2\right)=\left(4,\:8\right)[/tex]

As we know that the parallel lines have the same slopes.

Thus, the second step we need to do is to substitute m = 9/4 in the slope formula

[tex]\frac{9}{4}=\frac{y_2-y_1}{x_2-x_1}[/tex]

now substituting (x₁, y₁) = (-3, k) and (x₂, y₂) = (4, 8) in the formula

[tex]\:\:\:\frac{9}{4}=\:\frac{8-k}{4-\left(-3\right)}[/tex]

switch sides

[tex]\frac{8-k}{4-\left(-3\right)}=\frac{9}{4}[/tex]

Multiply both sides by 7

[tex]\frac{7\left(8-k\right)}{4-\left(-3\right)}=\frac{9\cdot \:7}{4}[/tex]

Simplify

[tex]8-k=\frac{63}{4}[/tex]

Subtract 8 from both sides

[tex]8-k-8=\frac{63}{4}-8[/tex]

Simplify

[tex]-k=\frac{31}{4}[/tex]

Divide both sides by -1

[tex]\frac{-k}{-1}=\frac{\frac{31}{4}}{-1}[/tex]

Simplify

[tex]k=-\frac{31}{4}[/tex]

Therefore, the value of k is:

[tex]k=-\frac{31}{4}[/tex]

VERIFICATION:

As the value of k = -31/4.

So the first line will contain the points:

(-3, -31/4)

(4, 8)

Finding the slope between (-3, -31/4) and (4, 8) using the slope formula

[tex]\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]

Here:

[tex]\left(x_1,\:y_1\right)=\left(-3,\:-\frac{31}{4}\right)[/tex]

[tex]\left(x_2,\:y_2\right)=\left(4,\:8\right)[/tex]

so substituting (x₁, y₁) = (-3, -31/4) and (x₂, y₂) = (4, 8) in the formula

[tex]m=\frac{8-\left(-\frac{31}{4}\right)}{4-\left(-3\right)}[/tex]

Apply rule: [tex]-\left(-a\right)=a[/tex]

[tex]m=\frac{8+\frac{31}{4}}{4+3}[/tex]

Add the numbers: [tex]4+3=7[/tex]

[tex]m=\frac{8+\frac{31}{4}}{7}[/tex]

[tex]m=\frac{\frac{63}{4}}{7}[/tex] ∵ [tex]8+\frac{31}{4}=\frac{63}{4}[/tex]

Apply fraction rule: [tex]\frac{\frac{b}{c}}{a}=\frac{b}{c\:\cdot \:a}[/tex]

[tex]m=\frac{63}{4\cdot \:7}[/tex]

[tex]m=\frac{63}{28}[/tex]

Cancel the common factor: 7

[tex]m=\frac{9}{4}[/tex]

So, the slope of the line containing the points (-3,k) and (4,8) is m = 9/4. It means both lines have the same slopes. Hence, they are parallel.

Therefore, we conclude that the value of k = -31/4 is true.