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Algebra / Linear equations in two variables Difficulty: Hard

$\frac{3}{5} x + \frac{3}{4} y = 7$

Which table gives three values of $x$ and their corresponding values of $y$ for the given equation?

$x$	$y$
$1$	$StartFraction 113 Over 20 EndFraction$
$2$	$StartFraction 101 Over 20 EndFraction$
$4$	$StartFraction 77 Over 20 EndFraction$

$x$	$y$
$1$	$StartFraction 47 Over 5 EndFraction$
$2$	$StartFraction 44 Over 5 EndFraction$
$4$	$StartFraction 38 Over 5 EndFraction$

$x$	$y$
$1$	$StartFraction 148 Over 15 EndFraction$
$2$	$StartFraction 136 Over 15 EndFraction$
$4$	$StartFraction 112 Over 15 EndFraction$

$x$	$y$
$1$	$StartFraction 128 Over 15 EndFraction$
$2$	$StartFraction 116 Over 15 EndFraction$
$4$	$StartFraction 92 Over 15 EndFraction$

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Explanation

Choice D is correct. Each of the tables gives the same three values of $x$ : $1$ , $2$ , and $4$ . Substituting $1$ for $x$ in the given equation yields $(\frac{3}{5}) (1) + \frac{3}{4} y = 7$ , or $\frac{3}{5} + \frac{3}{4} y = \frac{35}{5}$ . Subtracting $\frac{3}{5}$ from both sides of this equation yields $\frac{3}{4} y = \frac{32}{5}$ . Multiplying both sides of this equation by $\frac{4}{3}$ yields $y = \frac{128}{15}$ . Therefore, when $x equals 1$ , the corresponding value of $y$ for the given equation is $StartFraction 128 Over 15 EndFraction$ . Substituting $2$ for $x$ in the given equation yields $(\frac{3}{5}) (2) + \frac{3}{4} y = 7$ , or $\frac{6}{5} + \frac{3}{4} y = \frac{35}{5}$ . Subtracting $\frac{6}{5}$ from both sides of this equation yields $\frac{3}{4} y = \frac{29}{5}$ . Multiplying both sides of this equation by $\frac{4}{3}$ yields $y = \frac{116}{15}$ . Therefore, when $x equals 2$ , the corresponding value of $y$ for the given equation is $StartFraction 116 Over 15 EndFraction$ . Substituting $4$ for $x$ in the given equation yields $(\frac{3}{5}) (4) + \frac{3}{4} y = 7$ , or $\frac{12}{5} + \frac{3}{4} y = \frac{35}{5}$ . Subtracting $StartFraction 12 Over 5 EndFraction$ from both sides of this equation yields $\frac{3}{4} y = \frac{23}{5}$ . Multiplying both sides of this equation by $\frac{4}{3}$ yields $y = \frac{92}{15}$ . Therefore, when $x equals 4$ , the corresponding value of $y$ for the given equation is $StartFraction 92 Over 15 EndFraction$ . The table in choice D gives x-values of $1$ , $2$ , and $4$ and corresponding y-values of $StartFraction 128 Over 15 EndFraction$ , $StartFraction 116 Over 15 EndFraction$ , and $StartFraction 92 Over 15 EndFraction$ , respectively. Therefore, the table in choice D gives three values of $x$ and their corresponding values of $y$ for the given equation.

Choice A is incorrect. This table gives three values of $x$ and their corresponding values of $y$ for the equation $\frac{3}{5} x + \frac{3}{4} + y = 7$ .

Choice B is incorrect. This table gives three values of $x$ and their corresponding values of $y$ for the equation $\frac{3}{5} x + y = 10$ .

Choice C is incorrect. This table gives three values of $x$ and their corresponding values of $y$ for the equation $\frac{3}{5} x + \frac{3}{4} y = 8$ .