Math

QuestionSolve the system of equations: 2x15y=452x - \frac{1}{5}y = \frac{4}{5} and 13x12y=12\frac{1}{3}x - \frac{1}{2}y = -12. Find xx and yy.

Studdy Solution

STEP 1

Assumptions1. We have a system of two linear equations in two variables, x and y. . The system is consistent and has a unique solution.

STEP 2

The system of equations is given as2x15y=451x12y=12\begin{array}{l} 2 x-\frac{1}{5} y=\frac{4}{5} \\ \frac{1}{} x-\frac{1}{2} y=-12\end{array} We can solve this system using the method of substitution or elimination. Here, we will use the method of elimination.

STEP 3

First, we need to make the coefficients of y the same in both equations. To do this, we can multiply the first equation by10 and the second equation by6.
20x2y=82x3y=72\begin{array}{l} 20 x-2 y=8 \\ 2 x-3 y=-72\end{array}

STEP 4

Now, we can subtract the second equation from the first to eliminate y.
(20x2y)(2x3y)=8(72)\begin{array}{l} (20 x-2 y)-(2 x-3 y)=8-(-72) \end{array}

STEP 5

implify the equation to find the value of x.
18x+y=80\begin{array}{l} 18 x+y=80 \\ \end{array}

STEP 6

olve the equation for x.
x=80y18\begin{array}{l} x=\frac{80-y}{18} \end{array}

STEP 7

Now, substitute x into the second equation of the original system to solve for y.
13(80y18)12y=12\begin{array}{l} \frac{1}{3} (\frac{80-y}{18})-\frac{1}{2} y=-12\end{array}

STEP 8

implify the equation to find the value of y.
803y5412y=12\begin{array}{l} \frac{80-3y}{54}-\frac{1}{2} y=-12\end{array}

STEP 9

olve the equation for y.
y=80+10854+3=11657\begin{array}{l} y=\frac{80+108}{54+3}=\frac{116}{57} \end{array}

STEP 10

Substitute y into the equation for x to find the value of x.
x=8011605718=34057\begin{array}{l} x=\frac{80-\frac{1160}{57}}{18}=\frac{340}{57} \end{array} So, the solution to the system of equations isx=34057y=116057\begin{array}{l} x=\frac{340}{57} \\ y=\frac{1160}{57} \end{array}

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