Super-Hard SAT Math Question 134 | Free SAT Prep | Aniko

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Question 134·200 Super-Hard SAT Math Questions·Algebra

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\frac{5}{8}x - \frac{3}{4}y = m

\frac{15}{16}x - \frac{9}{8}y = 2n

The system of equations above has infinitely many solutions. What is the value of $\frac{m}{n}$ ? Anikо АI Tu‌tor

When a system has "infinitely many solutions," immediately recognize that one equation is a scalar multiple of the other. Find the multiplier $k$ by dividing corresponding coefficients (use $x$ or $y$ ). Then apply this same $k$ to the constant terms. Watch for extra factors like the $2$ in $2n$ —these change how you isolate the requested ratio. ЅАT рrep by Аnі‍ko.аі

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Recognize the condition

When a system has infinitely many solutions, both equations describe the same line. This means one equation is a scalar multiple of the other.

Find the multiplier

Compare corresponding coefficients between the two equations. Divide the second equation's $x$ -coefficient by the first equation's $x$ -coefficient to find the scale factor. Аniко.аі - Ѕ‍AT‍ Prep

Apply to the constants

The same scale factor must relate the right-hand sides. Set up the relationship between $m$ and $n$ using this factor, then solve for $\frac{m}{n}$ .

Desmos Guide

Compute the scale factor

In Desmos, enter:

k = (15/16) / (5/8)

Note the displayed value of $k$ . Anіk⁠о АІ⁠ Tuto⁠r

Set up the constant relationship

Since $2n = k \cdot m$ , enter:

m_over_n = 2/k

This represents $\frac{m}{n} = \frac{2}{k}$ .

Read the answer

Click the value for m_over_n and use exact fraction display to confirm the ratio.

Step-by-step Explanation

Identify the condition for infinitely many solutions

For a system to have infinitely many solutions, one equation must be a constant multiple $k$ of the other. All coefficients and constants must scale by the same factor.

Calculate the scale factor from the $x$ -coefficients

Compare the $x$ -coefficients: Anі⁠к‍o - Fr‍еe ‌ЅАT⁠ Рrеp

\begin{aligned} k &= \frac{\frac{15}{16}}{\frac{5}{8}} \\ &= \frac{15}{16} \cdot \frac{8}{5} \\ &= \frac{15 \cdot 8}{16 \cdot 5} \\ &= \frac{120}{80} \\ &= \frac{3}{2} \end{aligned}

Verify with the $y$ -coefficients

Check that the $y$ -coefficients also have ratio $\frac{3}{2}$ :

\frac{-\frac{9}{8}}{-\frac{3}{4}} = \frac{9}{8} \cdot \frac{4}{3} = \frac{36}{24} = \frac{3}{2} \checkmark

Apply the scale factor to the constants

Since the second equation equals $k$ times the first:

2n = k \cdot m = \frac{3}{2}m

Solve for $\frac{m}{n}$ :

\frac{m}{n} = \frac{2}{\frac{3}{2}} = 2 \cdot \frac{2}{3} = \frac{4}{3}

The answer is $\frac{4}{3}$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation