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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 4: Q. 345 (page 479)

In the following exercises, solve the systems of equations by elimination.
$\{\begin{cases} \frac{1}{3} x - \frac{1}{2} y = 1 \\ \frac{3}{4} x - y = \frac{5}{2} \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is $(6, 2)$ .

Step by step solution

01

Step 1. Given information.

The system of linear equations,

$\{\begin{cases} \frac{1}{3} x - \frac{1}{2} y = 1 \\ \frac{3}{4} x - y = \frac{5}{2} \end{cases}$

02

Step 2. Calculating the value of x.

Now, we will make some changes in the equations to eliminate $y$ terms.

Multiplying the first equation with $2$ we get

$2 (\frac{1}{3} x - \frac{1}{2} y = 1) \frac{2}{3} x - y = 2$

Then the equations will be,

$\frac{2}{3} x - y = 2 \frac{3}{4} x - y = \frac{5}{2}$

Subtracting the above equations then

$\frac{2}{3} x - y = 2 - \frac{3}{4} x + y = - \frac{5}{2} (\frac{2}{3} - \frac{3}{4}) x = 2 - \frac{5}{2} \frac{- 1}{12} x = \frac{- 1}{2} x = 6$

03

Step 3. Calculating the value of y.

Substituting the value of $x$ in any equation.

$\frac{1}{3} (6) - \frac{1}{2} y = 1 2 - \frac{1}{2} y = 1 - \frac{1}{2} y = - 1 y = 2$

04

Step 4. Checking the result.

Substituting the value of $x$ and $y$ in the first equation then

$\frac{1}{3} x - \frac{1}{2} y = 1 \frac{1}{3} (6) - \frac{1}{2} (2) = 1 2 - 1 = 1 1 = 1$

Substituting the value of $x$ and $y$ in the second equation then

$\frac{3}{4} x - y = \frac{5}{2} \frac{3}{4} (6) - 2 = \frac{5}{2} \frac{9}{2} - 2 = \frac{5}{2} 9 - 4 = 5 5 = 5$

Hence, this is true.

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