Chapter ISG, Problem 6.4.1P

a.

To determine

To describe two ways to solve the system by elimination.

Answer to Problem 6.4.1P

Addition or subtraction.

multiplication

Explanation of Solution

Given:

  $\begin{array}{l}-2x+3y=-5\\ 3x-4y=6\end{array}$

The two different ways of solving by elimination method is

• By addition or subtraction
• By multiplication

In addition, the coefficients of atleast 1 variable is opposite. Both the equations are added and one variable is wiped out. We substitute the value in equations after solving for another variable.

In multiplication, the equation is multiplied by a constant until the coefficient of at least 1 variable become opposite. Then the same procedure is followed as addition.

b.

To determine

To explain the incorrect.

Answer to Problem 6.4.1P

The answer is followed.

Explanation of Solution

Given:

  $\begin{array}{l}-2x+3y=-5\\ 3x-4y=6\end{array}$

As you know, the coefficient of the variable is the one to be made opposite, not the constant. Here, the multiplication of 1^st equation by 6 and second equation by 5 will make the constant equal but not the variables. Hence, it is not a useful measure to solve the system.

c.

To determine

To solve the system of equation

Answer to Problem 6.4.1P

  $x=-2$ and $y=-3$ is the solution

Explanation of Solution

Given:

  $\begin{array}{l}-2x+3y=-5\\ 3x-4y=6\end{array}$

Calculation:

Let’s assume equations be (1) and (2) respectively

To solve equations by elimination, the given equations are either added or subtracted to get an equation in only one variable.

Let’s eliminate variable x here. To eliminate a variable, the coefficients of the variable must be same in both the equations.

In equation 1, coefficient of x is -2 & in equation 2, coefficient of x is 3. To make the coefficients same, lets multiply equation 1 with $\frac{3}{2}$ .

  $\frac{3}{2}\times (-2x+3y)=-5\times \frac{3}{2}$

This gives us,

  $-3x+\frac{9}{2}y=\frac{-15}{2}$   (3)

Now, add the equation (3) and (2)

  $3x-4y-3x+\frac{9}{2}y=6-\frac{15}{2}$

Solving,$-4y+\frac{9}{2}y=6-\frac{15}{2}$

Now taking LCM,

  $\frac{-8y+9y}{2}=\frac{12-15}{2}$

Dividing and solving,

  $y=-3$

Put value of y in (1)

  $-2x+3(-3)=-5$

Solving,

  $-2x-9=-5$

So,

  $-2x=4$

Dividing by -2,

  $x=-2$

Hence, $x=-2$ and $y=-3$ is the solution

d.

To determine

Explanation of Solution

Given:

  $\begin{array}{l}-2x+3y=-5\\ 3x-4y=6\end{array}$

Graph:

The red line represents: $-2x+3y=-5$

The blue line represents: $3x-4y=6$

  

Interpretation:

Here, it can be seen that the intersection point is (-2, -3). This is also the solution of the graph.

Hence, the solution is correct.