Chapter 11.1, Problem 83AYU

To determine

To explain:The strategy to solve system of two linear equations containing two variables.

Answer to Problem 83AYU

The strategy to solve system of two linear equations containing two variables is either by substitution or elimination method.

Explanation of Solution

Given information:

The system of two linear equations containing two variables.

Calculation:

When there is asystem of two linear equations containing two variables the strategy used is either the method of elimination or substitution.

In elimination, one of the variable in both the equation is made same either by multiplying a number or dividing by a number. Then the equations are either added or subtracted to eliminate one variable. Now the value obtained for second variable is put in any of the equation to find the value of other variable.

In substitution, value of one variable is isolated from one equation and substituted in other and equations are simultaneously solved.

Consider the system of equations

  $\begin{array}{c}x+2y=7\\ y=5x-2\end{array}$

Multiply the second equation by 2,

  $\begin{array}{c}2y=10x-2\\ 10x-2y=4\end{array}$

Rewrite the system as,

  $\begin{array}{c}x+2y=7\\ 10x-2y=4\end{array}$

Use the method of elimination to solve the system above.

Add both the equations,

  $\begin{array}{ccccc}x& +& 2y& =& 7\\ 10x& -& 2y& =& 4\\ +& & & & \\ 11x& +& 0& =& 11\end{array}$

Therefore,

  $\begin{array}{c}11x=11\\ x=1\end{array}$

Now substitute $x=1$ in the equation $x+2y=7$ ,

  $\begin{array}{c}1+2y=7\\ 2y=6\\ y=3\end{array}$

Therefore, solution is the coordinate pair, $\left(1,3\right)$ .

Consider the system of equations

  $\begin{array}{c}x+y=6\\ 3x-2y=-2\end{array}$

Use the method of substitution to solve the system above.

From the equation $x+y=6$ isolate the value of y ,

  $\begin{array}{c}x+y=6\\ y=6-x\end{array}$

Now substitute $y=6-x$ in the equation $3x-2y=-2$ ,

  $\begin{array}{c}3x-2\left(6-x\right)=-2\\ 3x-12+2x=-2\\ 5x=-2+12\end{array}$

Simplify it further as,

  $\begin{array}{c}5x=10\\ x=2\end{array}$

Now, substitute $x=2$ in the equation $y=6-x$ ,

  $\begin{array}{c}y=6-2\\ y=4\end{array}$

Therefore, solution is the coordinate pair, $\left(2,4\right)$ .

Thus, the strategy to solve system of two linear equations containing two variables is either by substitution or elimination method.