The graph of a linear equation passes through the points and . The graph also passes through the point . Which choice is the value of ?

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Super-Hard SAT Math Question 95 | Free SAT Prep | Aniko

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Question 95·200 Super-Hard SAT Math Questions·Algebra

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The graph of a linear equation passes through the points $\left(-9, \frac{5}{4}\right)$ and $\left(15, -\frac{7}{2}\right)$ . The graph also passes through the point $\left(\frac{3}{2}, a\right)$ . Which choice is the value of $a$ ?

When a line is defined by two points and you need the y-value at a third x-value, the fastest path is: compute the slope with the fraction slope formula, write point-slope form using whichever point makes subtraction easiest, then substitute the given x. To avoid fraction mistakes, convert mixed steps like $\frac{3}{2}+9$ into a single fraction before multiplying by the slope, and use a common denominator for the final addition/subtraction.

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Start with the slope

Use the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ with the two given points. Keep the $y$ -values as fractions.

Use point-slope form

After you find $m$ , write an equation like $y-y_1=m(x-x_1)$ using one of the points.

Be careful with $x+9$

When substituting $x=\frac{3}{2}$ into $x-(-9)$ , rewrite it as $\frac{3}{2}+9=\frac{21}{2}$ before multiplying by the slope.

Desmos Guide

Enter the two points

Type the points as

A=(-9,5/4)
B=(15,-7/2)

Compute the slope from the points

Type m=(B.y-A.y)/(B.x-A.x) and let Desmos calculate the value.

Define the line using point-slope form

Type y=m(x-A.x)+A.y to graph the line through point A with slope m.

Evaluate the y-value when $x=\frac{3}{2}$

Type a=m*(3/2-A.x)+A.y. The value shown for a should match exactly one of the answer choices.

Step-by-step Explanation

Compute the slope

Using $\left(-9, \frac{5}{4}\right)$ and $\left(15, -\frac{7}{2}\right)$ ,

\begin{aligned} m &= \frac{-\frac{7}{2}-\frac{5}{4}}{15-(-9)} \\ &= \frac{-\frac{14}{4}-\frac{5}{4}}{24} \\ &= \frac{-\frac{19}{4}}{24} \\ &= -\frac{19}{96} \end{aligned}

Use point-slope form

Use the point $\left(-9, \frac{5}{4}\right)$ :

y-\frac{5}{4}=-\frac{19}{96}(x+9)

Substitute $x=\frac{3}{2}$

Plug in $x=\frac{3}{2}$ :

\begin{aligned} a-\frac{5}{4} &= -\frac{19}{96}\left(\frac{3}{2}+9\right) \\ &= -\frac{19}{96}\left(\frac{3}{2}+\frac{18}{2}\right) \\ &= -\frac{19}{96}\cdot\frac{21}{2} \end{aligned}

Finish the arithmetic

\begin{aligned} -\frac{19}{96}\cdot\frac{21}{2} &= -\frac{399}{192}=-\frac{133}{64} \\ \Rightarrow a &= \frac{5}{4}-\frac{133}{64} \\ &= \frac{80}{64}-\frac{133}{64}=-\frac{53}{64} \end{aligned}

So, $a$ is $-\frac{53}{64}$ .

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