In the xy-plane, a system of two linear equations has no solutions. One of the equations i… — SAT

In the $xy$ -plane, a system of two linear equations has no solutions. One of the equations in the system is $4x - 6y = 15$ . The graph of the second equation is a line that passes through the points $(3, k)$ and $(9, 7)$ , where $k$ is a constant. What is the value of $k$ ?

Cevap: 3

Cevap

The value of

k

3

For a system of linear equations to have no solutions, the lines representing the equations must be parallel, which means they have equal slopes but different y-intercepts. The first equation,

4x - 6y = 15

, can be rewritten in slope-intercept form as

y = \frac{2}{3}x - 2.5

, showing its slope is

\frac{2}{3}

. The slope of the second line, passing through

(3, k)

and

(9, 7)

, is given by

\frac{7 - k}{9 - 3} = \frac{7 - k}{6}

. Setting the two slopes equal yields

\frac{2}{3} = \frac{7 - k}{6}

. Multiplying both sides by

6

gives

4 = 7 - k

, which solves to

k = 3

. Substituting

k = 3

back into the second line gives

y = \frac{2}{3}x + 1

. Since the slopes are equal and the y-intercepts (

-2.5

and

1

) are different, the lines are parallel and distinct, confirming there are no solutions.

Adım Adım Çözüm

Find the slope of the first line by converting the equation to slope-intercept form.

Slope is

\frac{2}{3}

and y-intercept is

-2.5

Converting

4x - 6y = 15

y = \frac{2}{3}x - 2.5

reveals the slope of the first line.

Express the slope of the second line using the coordinates of the two points on the line.

Slope expression is

\frac{7 - k}{6}

Applying the slope formula

\frac{y_2 - y_1}{x_2 - x_1}

(3, k)

and

(9, 7)

defines the slope in terms of

k

Equate the two slopes and solve for the unknown parameter.

k = 3

Since the system has no solutions, the lines must be parallel and have equal slopes. Setting

\frac{2}{3} = \frac{7 - k}{6}

and solving gives

k = 3

Anahtar Kavram

For a system of two linear equations to have no solutions, the lines representing the equations must be parallel, which requires them to have the same slope but different y-intercepts.

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