How do I use elimination to find the solution of the system of equations $y = \frac{1}{3} x + \frac{7}{3}$ $y=1/3x+7/3$ and $y = - \frac{5}{4} x + \frac{11}{4}$ $y=−5/4x+11/4$ ?

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How do I use elimination to find the solution of the system of equations $y = \frac{1}{3} x + \frac{7}{3}$ $y=1/3x+7/3$ and $y = - \frac{5}{4} x + \frac{11}{4}$ $y=−5/4x+11/4$ ?

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Answer 1 · 2015-07-09T20:54:34

You multiply each equation by the LCM of the denominators and then solve by elimination as usual.

Explanation:

(1) $y = \frac{1}{3} x + \frac{7}{3}$ $y = 1/3x+7/3$
(2) $y = - \frac{5}{4} x + \frac{11}{4}$ $y = -5/4x + 11/4$

Step 1: Multiply each equation by the least common multiple (LCM) of the denominator.

Multiply Equation 1 by $3$ $3$ and Equation 2 by $4$ $4$ .

(3) $3 y = x + 7$ $3y = x+7$
(4) $4 y = - 5 x + 11$ $4y = -5x + 11$

Step 2: To solve by elimination, multiply Equation 3 by $5$ $5$ .

(5) $15 y = 5 x + 35$ $15y = 5x+35$
(4) $4 y = - 5 x + 11$ $4y = -5x + 11$

Step 3: Add the two equations.

$19 y = 46$ $19y = 46$

(6) $y = \frac{46}{19}$ $y= 46/19$

Step 4: Substitute Equation 6 into Equation 3.

$x + 7 = 3 y = 3 \times \frac{46}{19} = \frac{138}{19}$ $x+7 = 3y = 3×46/19 = 138/19$

$x = \frac{138}{19} - 7 = \frac{138 - 133}{19} = \frac{5}{19}$ $x= 138/19 -7 = (138-133)/19 = 5/19$

The solution is $x = \frac{5}{19}$ $x = 5/19$ , $y = \frac{46}{19}$ $y = 46/19$

Check:

Substitute in Equation 1.

$y = \frac{1}{3} x + \frac{7}{3}$ $y = 1/3x+7/3$

$\frac{46}{19} = \frac{1}{3} \times \frac{5}{19} + \frac{7}{3} = \frac{5}{57} + \frac{7}{3} = \frac{5}{57} + \frac{133}{57} = \frac{138}{57} = \frac{46}{19}$ $46/19 = 1/3× 5/19 + 7/3 = 5/57 + 7/3 = 5/57 + 133/57 =138/57 =46/19$

Substitute in Equation 2.

$y = - \frac{5}{4} x + \frac{11}{4}$ $y = -5/4x + 11/4$

$\frac{46}{19} = - \frac{5}{4} \times \frac{5}{19} + \frac{11}{4} = - \frac{25}{76} + \frac{11}{4} = - \frac{25}{76} + \frac{209}{76} = \frac{184}{76} = \frac{46}{19}$ $46/19 = -5/4× 5/19 + 11/4 = -25/76 + 11/4 = -25/76 + 209/76 = 184/76 = 46/19$

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How do I use elimination to find the solution of the system of equations $y = \frac{1}{3} x + \frac{7}{3}$ $y=1/3x+7/3$ and $y = - \frac{5}{4} x + \frac{11}{4}$ $y=−5/4x+11/4$ ?

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How do I use elimination to find the solution of the system of equations $y = \frac{1}{3} x + \frac{7}{3}$ $y=1/3x+7/3$ and $y = - \frac{5}{4} x + \frac{11}{4}$ $y=−5/4x+11/4$ ?