Chapter 10, Problem 1RCC

1. (a) What are the three methods we use to solve a system of equations?
2. (b) Solve the system by the elimination method and by the graphical method.

$\{\begin{array}{r}\hfill x+y=3\\ \hfill 3x-y=1\end{array}$

To determine

The three methods used to solve a system of equations.

Answer to Problem 1RCC

The three methods to solve a system of equations are Substitution Method, Elimination Method and Graphical Method.

Explanation of Solution

The first method is Substitution Method.

Solve any of the two given equations in terms of the other variable and substitute the value into the other equation then back substitute the value into the initial equation to determine the value of the variables.

The second method is Elimination Method.

Multiply the equations with a selected constant value to make coefficient of any of the two variable same with opposite sign and add the equations to get the value of one of the variables then back substitute the value into the initial equation to determine the value of the other variable.

The third method is Graphical Method.

Graph the given pair of equations and find the intersection point of the graphs . The x and y coordinates of the intersection point is the value of variables.

To determine

To evaluate: The solution of given equations using elimination method and graphical method.

Answer to Problem 1RCC

The solution is ordered pair $\left(1,2\right)$.

Explanation of Solution

Section1:

The given equations are,

$x+y=3$ (1)

$3x-y=1$ (2)

Add the equation (1) and (2) to eliminate y, as the coefficients of y-term in both equations are negative to each other.

$\begin{array}{r}x+y=3\\ \underset{\_}{3x-y=1}\\ 4x=4\\ x=1\end{array}$

The value of x is 1.

Back-substitute 1 for x in equation (1) and find the value of y.

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

The value of y is 2.

Thus, the solution is ordered pair $\left(1,2\right)$.

Section2

The given equations are,

$x+y=3$ (1)

$3x-y=1$ (2)

Substitute some value of y in equation (1) and make a table for values of x and y.

x	y
$-1$	4
0	3
1	2
2	1
3	0

Substitute some value of y in equation (2) and make a table for values of x and y.

x	y
$-1$	$-4$
0	$-1$
1	2
2	5
3	8

Plot the point from the tables and connect the points to make the graph of equation (1) and equation (2).

Figure (1)

From Figure (1), it can be noticed that the graph of two linear equations intersect at a single point.

Thus, the system of equations has one solution and the point of intersection is $\left(1,2\right)$.