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Exercises Systems of equations

Example 7 from Algebra and Trigonometry, 11.1 Systems of Linear Equations: Two Variables

Using the Addition Method in Systems of Equations Containing Fractions

Solve the given system of equations in two variables by addition.

\begin{array}{l} \frac{x}{3} + \frac{y}{6} = 3 \\ \frac{x}{2} - \frac{y}{4} = ​ 1 \end{array}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
First clear each equation of fractions by multiplying both sides of the equation by the least common denominator.

$\begin{array}{l} 6 (\frac{x}{3} + \frac{y}{6}) = 6 (3) \\ 2 x + y = 18 \\ 4 (\frac{x}{2} - \frac{y}{4}) = 4 (1) \\ 2 x - y = 4 \end{array}$
Now multiply the second equation by $−1$ so that we can eliminate the x-variable.
$\begin{array}{l} −1 (2 x - y) = −1 (4) \\ −2 x + y = −4 \end{array}$
Add the two equations to eliminate the x-variable and solve the resulting equation.
$\begin{array}{l} 2 x + y = 18 \\ \underset{_____________}{−2 x + y = −4} \\ 2 y = 14 \\ y = 7 \end{array}$
Substitute $y = 7$ into the first equation.

$\begin{array}{l} 2 x + (7) = 18 \\ 2 x = 11 \\ x = \frac{11}{2} \\ = 5.5 \end{array}$
The solution is $(\frac{11}{2}, 7) .$ Check it in the other equation.

$\begin{matrix} \frac{x}{2} - \frac{y}{4} = 1 \\ \frac{\frac{11}{2}}{2} - \frac{7}{4} = 1 \\ \frac{11}{4} - \frac{7}{4} = 1 \\ \frac{4}{4} = 1 \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve the system of equations by addition.

\begin{matrix} 2 x + 3 y & = & 8 \\ 3 x + 5 y & = & 10 \end{matrix}

How did it go?

PreviousUsing the Addition Method When Multiplication of Both Equations Is Required Next Solving an Inconsistent System of Equations

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 11.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.