Chapter 6.5, Problem 25HP

To determine

To find the solution of the given system of equations using three methods

Answer to Problem 25HP

(2, 5)

Explanation of Solution

Given:

System of equations:

  $\begin{array}{l}4x+y=13\\ 6x-y=7\end{array}$

Concept used:

Three different methods of solving the system of equations:

• Substitution
• Elimination
• Graphing

Calculation:

Substitution:

  $\begin{array}{l}\Rightarrow 4x+y=13\\ \Rightarrow y=13-4x\mathrm{..........}(1)\end{array}$

  $6x-y=\mathrm{7................}(2)$

Substitute equation (1) in equation (2).

  $\begin{array}{l}\Rightarrow 6x-(13-4x)=7\\ \Rightarrow 6x-13+4x=7\\ \Rightarrow 10x=20\\ \Rightarrow x=2\end{array}$

Substitute the value of x in equation (1).

  $\begin{array}{l}\Rightarrow y=13-4(2)\\ \Rightarrow y=13-8\\ \Rightarrow y=5\end{array}$

Elimination:

  $\begin{array}{l}4x+y=\mathrm{13........................}(3)\\ 6x-y=\mathrm{7..........................}(4)\end{array}$

Add equation (3) and equation (4), so that y is eliminated.

  $\begin{array}{l}\Rightarrow (4x+y)+(6x-y)=13+7\\ \Rightarrow 10x=20\\ \Rightarrow x=2\end{array}$

Substitute the value of x in equation (3).

  $\begin{array}{l}\Rightarrow 4(2)+y=13\\ \Rightarrow 8+y=13\\ \Rightarrow y=5\end{array}$

Graphing:

Write the system of equations in slope-intercept form ( $y=mx+c$ ).

  $\begin{array}{l}4x+y=13\\ \Rightarrow y=-4x+\mathrm{13...........}(1)\\ 6x-y=7\\ \Rightarrow y=6x-\mathrm{7...............}(2)\end{array}$

Substitute random values for x and find y as follows $y=-4x+13$

x	-2	-1	0	1	2
y	21	17	13	9	5

  $y=6x-7$

x	-2	-1	0	1	2
y	-19	-13	-7	-1	5

Plot the points in a graph. The point of intersection of the two lines is the solution to the system of equations.

Graph:

In the below graph, the blue line represents (1) and the green line represents (2)

  

Inference:

The point of intersection of both the lines is (2, 5)

Conclusion:

Therefore, the solution of the system of equations is (2, 5) in all the three methods.