Consider the system of linear equations below: \begin{aligned} 3x + 4y &= 8 \\ ax + 8y &= … — SAT

Consider the system of linear equations below:

\begin{aligned} 3x + 4y &= 8 \\ ax + 8y &= 15 \end{aligned}

If the system of equations has no solution, and $a$ is a constant, what is the value of $a$ ?

Answer: 6

Answer

The correct answer is 6.

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

3x + 4y = 8

in slope-intercept form gives

y = -\frac{3}{4}x + 2

, so its slope is

-\frac{3}{4}

. Writing

ax + 8y = 15

in slope-intercept form gives

y = -\frac{a}{8}x + \frac{15}{8}

, so its slope is

-\frac{a}{8}

. Equating the slopes gives

-\frac{3}{4} = -\frac{a}{8}

. Solving for

a

gives

a = 6

. Since the y-intercepts (

2

and

\frac{15}{8}

) are different, the lines are parallel and have no intersection points.

Step-by-Step Solution

Convert the first equation

3x + 4y = 8

to slope-intercept form.

y = -\frac{3}{4}x + 2

To find the slope of the first line, which is

-\frac{3}{4}

Convert the second equation

ax + 8y = 15

to slope-intercept form.

y = -\frac{a}{8}x + \frac{15}{8}

To find the slope of the second line in terms of

a

, which is

-\frac{a}{8}

Equate the slopes of the two lines.

-\frac{3}{4} = -\frac{a}{8}

Parallel lines have the same slope, and a system with parallel lines has no solution.

Solve the equation for

a

a = 6

Isolating the variable

a

yields the value of the constant.

Key Concept

A system of linear equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

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