8x+5y=9 and 3x+2y=4 Create an equivalent system of equations

Question

A system of equations is shown below.

8x+5y=9 and 3x+2y=4

Part A: Create an equivalent system of equations by replacing on equation with the sum of that equation and a multiple of the other. Show the steps to do this.

Part B: Show that the equivalent system has the same solution as the original system of equations.

 

Answer 1

Part A :

If in a system, an equation is replaced by another equation that is obtained from adding the two equations from a system

previously multiplied or divided by a non zero number, the resultant system is equivalent.

The equations are

Multiply the equations by 2

Now the equivalent equations are

Multiply the third equation by 4.

Now the equation is
        Equation (3)

Multiply the fourth equation by negitive 10.

               Equation (2)

add the equations to eliminate variable y.

(Write the equations in column form and add)

Substitute in either equation to find the value of y.

(Equation 3)

The solution is .

Answer 2

continuous...

Part B

The equations are

Multiply the first equation by 2

Now the equation is

                Equation (1)

Multiply the second equation by negitive 5

Now the equation is

               Equation (2)

add the equations to eliminate variable x.

(Write the equations in column form and add)

Substitute in either equation to find the value of y.

    (Equation 1)

                 

                         

The solution is .

 The equivalent system has the same solution as the original system of equations.

Answer 3

Elimination Method :

The system of equations are .

(a) A system of linear equations is transformed into an equivalent system if :

1. Two equations are interchanged.
2. An equation is multiplied by a nonzero constant.
3. A constant multiple of another equation is added to a given equation.

Neither variable has a common coefficient.The coefficient  of the y - variables are 5 and 2 and their least common multiple is 10, so multiply each equation by the value  that will make the y - coefficient 10.

To get two equations that contain opposite terms multiply the first equation by negative 2 and the second equation by 5.

Write the equations in column form and add the corresponding columns to eliminate y - variable.

The resultant equation is .

The equation x = - 2 paired with either of the two original equations produces an equivalent system.So, substitute the value of x = - 2 in either of the original equations to solve for y.

The first equation: .

.

The solution of equivalent system (x, y) = (- 2, 5).

(b) Show that the equivalent system has the same solution as the original system of equations.

To check the solution, substitute the value of (x, y) = (- 2, 5) in the original system, as follows.

Therefore linear system and equivalent system has same solution.

The solution is .