Consider the system of linear equations where is a constant. For exactly one value of , the system has no solutions. What is that value of ?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 66 | Free SAT Prep | Aniko

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Question 66·Hard·Systems of Two Linear Equations in Two Variables

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Consider the system of linear equations

ax + 3y = 9

4x - 2y = 6,

where $a$ is a constant.

For exactly one value of $a$ , the system has no solutions. What is that value of $a$ ?

For systems of two linear equations asking about “no solutions” (or “infinitely many solutions”), think in terms of the graphs: two lines. Quickly rewrite both equations in slope-intercept form $y=mx+b$ , then compare their slopes and intercepts. Equal slopes with different intercepts mean no solutions (parallel distinct lines); equal slopes and equal intercepts mean infinitely many solutions (same line); different slopes mean exactly one solution. This approach avoids heavy algebra and lets you answer efficiently by focusing on slopes.

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Think about what “no solutions” means geometrically

If two linear equations in $x$ and $y$ have no solution, what must their graphs (lines) look like in the coordinate plane?

Compare slopes

Rewrite both equations in the form $y = mx + b$ so you can easily compare their slopes and y-intercepts.

Use the condition for parallel lines

Set the slopes of the two lines equal and solve for $a$ . Then make sure the y-intercepts are not the same so that the lines are parallel but not identical.

Desmos Guide

Enter the second equation in slope-intercept form

In Desmos, type y = 2x - 3 to graph the second equation (this comes from rewriting $4x - 2y = 6$ ).

Enter the first equation with a slider for a

Type y = (-a/3)x + 3. Desmos will automatically create a slider for a, allowing you to change its value and see how the line moves.

Adjust the slider until the lines are parallel

Move the a slider and watch how the first line’s tilt (slope) changes. Find the value of a where the two lines in the graph have exactly the same tilt (same slope) but cross the y-axis at different points.

Confirm the lines never intersect

Once the lines are parallel with different y-intercepts, check the a value shown on the slider. That is the value of $a$ for which the system has no solutions, because the lines never meet.

Step-by-step Explanation

Translate “no solutions” into a condition on the lines

For a system of two linear equations in $x$ and $y$ , no solutions means the lines never meet.

That happens exactly when the lines are parallel but not the same line, which in algebra means:

The slopes are equal, and
The y-intercepts are different.

Put both equations into slope-intercept form

Rewrite each equation as $y = mx + b$ to see the slopes clearly.

First equation:

Start with $ax + 3y = 9$ .
Solve for $y$ :

3y = -ax + 9 \\[0.7em] y = -\frac{a}{3}x + 3

So the slope of the first line is $m_1 = -\dfrac{a}{3}$ and the y-intercept is $3$ .

Second equation:

Start with $4x - 2y = 6$ .
Solve for $y$ :

-2y = -4x + 6 \\[0.7em] y = 2x - 3

So the slope of the second line is $m_2 = 2$ and the y-intercept is $-3$ .

Set the slopes equal to make the lines parallel

For the system to have no solutions, the slopes must be equal:

-\frac{a}{3} = 2

Solve this equation for $a$ :

- a = 6 \\[0.7em] a = -6

So when $a$ has this value, the two lines are parallel.

Check that the lines are not the same line

When $a=-6$ , the two equations become:

First line: $y = 2x + 3$ (since $-(-6)/3 = 2$ )
Second line: $y = 2x - 3$

They have the same slope $2$ but different y-intercepts ( $3$ vs. $-3$ ), so they are parallel and distinct, meaning the system has no solutions.

Therefore, the value of $a$ that makes the system have no solutions is $-6$ .

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Step-by-step Explanation

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Step-by-step Explanation