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Solve the system by elimination. Eliminate all denominators first for systems with fractions. 4x...

Question:

Solve the system by elimination. Eliminate all denominators first for systems with fractions.

{eq}4x \,+\, 3y \,=\, -1 \\ 2x \,+\, 5y \,=\, 3 {/eq}

Elimination Method:

  • Elimination method is one of the methods that is used to solve a system of equations.
  • In this method, we multiply one or both equations by some numbers such that the coefficients of one of the variables are the same but are of the opposite signs.
  • We then add the equations to get the eliminated variable.
  • Solve the resultant equation for the other variable.

Answer and Explanation:

The given two equations:

$$\begin{aligned} &4 x+3 y=-1&\rightarrow (1)\\ &2 x+5 y=3&\rightarrow (2) \end{aligned} $$

Multiply equation {eq}(2){/eq} both sides by {eq}-2 {/eq}:

$$\begin{align} &-4x-10y = -6 & \rightarrow (3) \end{align} $$

We add the equations {eq}(1){/eq} and {eq}(3){/eq}:

$$\begin{align} (4 x+3 y)+(-4x-10y) &=-1+(-6) \\[0.3cm] (4x-4y)+(3y-10y) &=-7 \\[0.3cm] -7y &=-7\\[0.3cm] y&=1 & \left[ \text{Divide both sides by -7}\right] \end{align} \\ $$

Substitute this in {eq}(2){/eq} (or in any other equation):

$$\begin{align} 2x+ 5(1) & = 3 \\[0.3cm] 2x+5 &= 3& \left[ \text{Subtracting 5 from both sides}\right]\\[0.3cm] 2x&=-2\\[0.3cm] x&=-1 & \left[ \text{Divide both sides by 2}\right] \end{align} \\ $$

Therefore, the solution is {eq}\color{blue}{\boxed{\mathbf{x=-1; \,\,y=1}}} {/eq}.


Learn more about this topic:

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Elimination Method in Algebra: Definition & Examples

from High School Algebra II: Help and Review

Chapter 7 / Lesson 9
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