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Exercises Systems of equations

Example 6 from Algebra and Trigonometry, 11.1 Systems of Linear Equations: Two Variables

Using the Addition Method When Multiplication of Both Equations Is Required

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

\begin{matrix} 2 x + 3 y = −16 \\ 5 x −10 y = 30 \end{matrix}

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
One equation has $2 x$ and the other has $5 x .$ The least common multiple is $10 x$ so we will have to multiply both equations by a constant in order to eliminate one variable. Let’s eliminate $x$ by multiplying the first equation by $−5$ and the second equation by $2.$
$\begin{array}{l} - 5 (2 x + 3 y) = - 5 (−16) \\ - 10 x - 15 y = 80 \\ 2 (5 x - 10 y) = 2 (30) \\ 10 x - 20 y = 60 \end{array}$
Then, we add the two equations together.

$\begin{array}{l} \underset{________________}{\begin{array}{l} \begin{array}{l} −10 x −15 y = 80 \\ 10 x −20 y = 60 \end{array} \end{array}} \\ −35 y = 140 \\ y = −4 \end{array}$
Substitute $y = −4$ into the original first equation.
$\begin{matrix} 2 x + 3 (−4) = −16 \\ 2 x - 12 = −16 \\ 2 x = −4 \\ x = −2 \end{matrix}$
The solution is $(−2, −4) .$ Check it in the other equation.
$\begin{array}{r} 5 x −10 y = 30 \\ 5 (−2) −10 (−4) = 30 \\ −10 + 40 = 30 \\ 30 = 30 \end{array}$
See Figure 7.

Figure 7

How did it go?

PreviousUsing the Addition Method When Multiplication of One Equation Is Required Next Using the Addition Method in Systems of Equations Containing Fractions

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.