Chapter 5.3, Problem 14E

To determine

To Solve: The given system of linear equations by elimination.

  $\begin{array}{l}10x-9y=46\\ -2x+3y=10\end{array}$

Answer to Problem 14E

The required solution is $x=2,y=1$

Explanation of Solution

Given information:

  $\begin{array}{l}10x-9y=46\\ -2x+3y=10\end{array}$

Calculation:

  $\begin{array}{l}10x-9y=46\text{ Equation 1}\\ -2x+3y=10\text{ Equation 2 }\end{array}$

Multiply Equation 2 by 3 so that the co-efficient of y-terms are opposites

  $\begin{array}{l}3\left(-2x+3y\right)=3\cdot 10\\ -6x+9y=30\text{ Revised Equation 2}\end{array}$

Variable y can be eliminated by adding equation 1 and revised equation 2

  $\begin{array}{l}10x-9y=46\\ \underset{\_}{-6x+9y=30}\\ \text{ 4}x=76\end{array}$

Divide 4 on both sides:

  $\begin{array}{l}\Rightarrow \frac{4x}{4}=\frac{76}{4}\\ \Rightarrow x=19\end{array}$

Now substitute the value of x in equation 1

  $\begin{array}{l}10x-9y=46\\ \Rightarrow 10\cdot 19-9y=46\\ \Rightarrow 190-9y=46\end{array}$

Subtract 190from both sides

  $\begin{array}{l}\Rightarrow 190-190-9y=46-190\\ \Rightarrow -9y=-144\end{array}$

Divide -9 on both sides

  $\begin{array}{l}\Rightarrow \frac{-9y}{-9}=\frac{-144}{-9}\\ \Rightarrow y=16\end{array}$

Check

Right hand side of equation 1 is 46

Left hand side of equation 1 is $10x-9y$

Substituting the known value and simplify

  $\begin{array}{l}=10\cdot 19-9\cdot 16\\ =190-144\\ =46\end{array}$

Left hand side = Right hand side.

Right hand side of equation 2 is 10

Left hand side of equation 2 is $-2x+3y$

Substituting the known value and simplify

  $\begin{array}{l}=-2\cdot 19+3\cdot 16\\ =-38+48\\ =10\end{array}$

Left hand side = Right hand side.