Solve these systems of equations using all three methods?

Question

1.) 3x + y = 5

x - 2y=4

2.) y - x = 5

2x - 2y=10

3.) 9x - 2y = 33

x - 6 = y

4.) 5x - 7y = -16

2x + 8y = 26

Please help don't understand how these are solved! Thank you

Answer 1

(1) The system of equations are 3x + y = 5 and x - 2y = 4.

To determine the best method to solve the system of equations, look closely at the coefficients of each term.

Since neither the coefficients of x nor the coefficients of y are the same or additive inverses, here cannot add or subtract to eliminate.

Since the coefficient of x in the second equation is 1, you can use the substitution method and also use elimination using multiplication.

Substitution Method :

Solve the first equation for y since the coefficient of y is 1 (or) Solve the second equation for x since the coefficient of x is 1.

Second equation : x - 2y = 4 -----> x = 4 + 2y.

Find the value of y by substituting 4+ 2y for x in the First equation.

First equation : 3x + y = 5

3(4 + 2y) + y = 5

12 + 6y + y = 5

12 + 7y = 5

7y = - 7

y = - 1.

Substitute - 1 for y in either equation to find the value of x.

Second equation : x - 2y = 4.

x - 2(- 1) = 4

x + 2 = 4

x = 2.

The solution (x, y) = (2, - 1).

Answer 2

(2) The system of equations are y - x = 5 and 2x - 2y = 10.

To determine the best method to solve the system of equations, look closely at the coefficients of each term.

Since neither the coefficients of x nor the coefficients of y are the same or additive inverses, here cannot add or subtract to eliminate.

Since the coefficient of x in the first equation is - 1, use the substitution method or elimination using multiplication.

Substitution Method :

Solve the first equation for y since the coefficient of y is 1.

First equation : y - x = 5 -----> y = 5 + x.

Find the value of x by substituting 5 + x for y in the second equation.

Second equation : 2x - 2y = 10.

2x - 2(5 + x) = 10

2x - 10 - 2x = 10

- 10 = 10.

The above statement is false, so the system equation has no solution.

Answer 3

(3) The system of equations are 9x - 2y = 33 and x - 6 = y.

To determine the best method to solve the system of equations, look closely at the coefficients of each term.

Since neither the coefficients of x nor the coefficients of y are the same or additive inverses, here cannot add or subtract to eliminate.

Since the coefficient of y in the second equation is 1 and it is also solved for y, use the substitution.

Substitution Method :

The second equation : y = x - 6 is already solved for y.

Find the value of x by substituting x - 6 for y in the First equation.

First equation : 9x - 2y = 33.

9x - 2(x - 6) = 33

9x - 2x + 12 = 33

7x + 12 = 33

7x = 21

x = 3.

Substitute 3 for x in either equation to find the value of y.

Second equation : y = x - 6.

y = (3) - 6

y = - 3

The solution (x, y) = (3, - 3).

 

Answer 4

(4) The system of equations are 5x - 7y = - 16 and 2x + 8y = 26.

To determine the best method to solve the system of equations, look closely at the coefficients of each term.

Elimination using multiplication :

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

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

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

The resultant equation is 54y = 162 ------> y = 3.

Substitute the value of y = 3 in either of the original equations and solve for x.

The first equation: 5x - 7y = - 16.

5x - 7(3) = - 16

5x - 21 = - 16

5x = 5

x = 1.

The solution (x, y) = (1, 3).