\begin{aligned} 3x - 5y &= 8 \\ kx + 10y &= -3 \end{aligned} In the system of equations ab… — SAT

\begin{aligned} 3x - 5y &= 8 \\ kx + 10y &= -3 \end{aligned}

In the system of equations above, $k$ is a constant. If the system has no solution, what is the value of $k$ ?

-6

The correct answer is

-6

. A system of two linear equations has no solution when the equations represent parallel lines, meaning they have the same slope but different

y

-intercepts. Writing the first equation in slope-intercept form gives

y = \frac{3}{5}x - \frac{8}{5}

, so its slope is

\frac{3}{5}

. Writing the second equation in slope-intercept form gives

y = -\frac{k}{10}x - \frac{3}{10}

, so its slope is

-\frac{k}{10}

. Since the

y

-intercepts are different (

-\frac{8}{5} \neq -\frac{3}{10}

), the system will have no solution when their slopes are equal:

\frac{3}{5} = -\frac{k}{10}

. Solving this equation for

k

yields

k = -6

Convert the first equation to slope-intercept form (

y = mx + b

) to find its slope.

y = \frac{3}{5}x - \frac{8}{5}

, so the slope is

\frac{3}{5}

and the

y

-intercept is

-\frac{8}{5}

Expressing the line in slope-intercept form directly reveals its slope and

y

-intercept.

Convert the second equation to slope-intercept form to express its slope in terms of

k

10y = -kx - 3 \Rightarrow y = -\frac{k}{10}x - \frac{3}{10}

, so the slope is

-\frac{k}{10}

and the

y

-intercept is

-\frac{3}{10}

This allows comparison of the slope and

y

-intercept of the second line with the first line.

Set the slopes equal to each other and solve for

k

\frac{3}{5} = -\frac{k}{10} \Rightarrow 30 = -5k \Rightarrow k = -6

A system of two linear equations has no solution if and only if the lines are parallel (equal slopes) and have different

y

-intercepts (which

-\frac{8}{5}

and

-\frac{3}{10}

are).

Determining the number of solutions to a system of linear equations based on slope and intercept relationships.

Soru