Question 98·200 Super-Hard SAT Math Questions·Algebra

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In the given system of equations, $k$ is a constant. Which choice gives the value of $k$ for which the system has no solution?

\frac{3}{5}x+\frac{2}{3}y=\frac{7}{10}-\frac{4}{3}y

\frac{k}{2}x+\frac{10}{3}y=\frac{9}{5}+\frac{1}{3}y

A linear system has no solution when its lines are parallel but distinct. Put both equations in $Ax+By=C$ form, with the variables on the left. Then match the variable coefficients, then check the constants. If the constants match too, the equations describe the same line instead. Desmos can confirm that the distinct lines never cross.

Hints

Hint 1
To compare the lines, first put each equation in standard form $Ax+By=C$ : move its $y$ terms to the left. What are the combined $y$ coefficients in the two equations?
Hint 2
For parallel lines, the $x$ and $y$ coefficients in one equation must both be the same multiple of those in the other. What multiplier takes the first equation's $y$ coefficient to the second's?
Hint 3
Matching variable coefficients isn't enough for no solution. If the constants match under that multiplier too, the equations draw the same line. Compare the constants before finding $k$ .

Step-by-step

Match coefficients, then check the lines
Step 1
Combine the first equation's $y$ terms
Add $\frac{4}{3}y$ to both sides to put the variables on the left:
$\frac{3}{5}x+\frac{2}{3}y+\frac{4}{3}y=\frac{7}{10}.$
Combine the $y$ terms:
$\frac{3}{5}x+2y=\frac{7}{10}.$
Step 2
Combine the second equation's $y$ terms
Subtract $\frac{1}{3}y$ from both sides:
$\frac{k}{2}x+\frac{10}{3}y-\frac{1}{3}y=\frac{9}{5}.$
Combine the $y$ terms:
$\frac{k}{2}x+3y=\frac{9}{5}.$
Step 3
Find the matching $x$ coefficient
The first line has $2y$ and the second has $3y$ , so multiply the first line's variable coefficients by $\frac{3}{2}$ . Type $\frac{3}{2}\cdot\frac{3}{5}$ in Desmos. It shows $0.9$ , or $\frac{9}{10}$ with the fraction button. For the lines to be parallel, the second line's $x$ coefficient must also be $\frac{9}{10}$ .
Step 4
Check that the constants do not match
Apply that same multiplier to the first line's constant. Type $\frac{3}{2}\cdot\frac{7}{10}$ ; Desmos shows $1.05$ , or $\frac{21}{20}$ with the fraction button. That is not the second line's constant, $\frac{9}{5}$ . So matching the variable coefficients will give two different lines, not one line written twice.
Step 5
See the parallel lines in Desmos
Graph $\frac{3}{5}x+2y=\frac{7}{10}$ and $\frac{9}{10}x+3y=\frac{9}{5}$ . Desmos shows separate parallel lines: they never cross, so this pair has no solution. The second graph is the given second equation once its $x$ coefficient, $\frac{k}{2}$ , equals $\frac{9}{10}$ .
Step 6
Find the value of $k$
Solve $\frac{k}{2}=\frac{9}{10}$ . Type $\frac{x_1}{2}\sim\frac{9}{10}$ : $x_1$ stands for the unknown $k$ , and $\sim$ tells Desmos to find its value rather than make a slider. Desmos reports $x_1=1.8$ under PARAMETERS. Type $x_1$ on the next line and use its fraction button to see $\frac{9}{5}$ . So $k=\frac{9}{5}$ makes the system have no solution. Choice D.

Question 98·200 Super-Hard SAT Math Questions·Algebra

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