Algebra 1 help? solving systems of linear equations?

Question

how to solve so if you can help step by step that would be great! 
1) 
2x+y≤8 
3x-y<2 

then there is a graph 

2) how are the graphs of systems of linear equations and inequalities related to their solutions 

3) Julie is solving the linear system below by substitution. 
2x + y = 7 
3x - 2y = -7 

which of the following would be a step in solving the system 
a) substitute y + 7 for x un 3x - 2y = -7 
b) substitute -y + 7 for x in 3x - 2y = -7 
c) substitute -2x + 7 for y in 3x - 2y = -7 
d) substitute 2x - 7 for y in 3x -2y = -7 

4) Marco and tam play video games a total of 15 hours each week. Marco plays 3 hours more than twice the number of hours tam plays. Write a system of linear equations to represent the situation. How many hours do each play? 

Answer 1

1)The inequalities are 2x + y ≤ 8 and 3x - y < 2.

• Draw the coordinate plane.

Now first inequality 2x + y ≤ 8.

•  Graph the line y = - 2x + 8
•  Since the inequality symbol is ≤  the boundary is included the solution set.

Graph the boundary of the inequality 2x + y ≤ 8 with solid line.

•  To determine which half plane to be shaded use a test point in either half- plane.

A simple choice is (0,0). Substitute x = 0 and y = 0 in original inequality

2(0) + 0 ≤ 8

0 ≤ 8

The statement is true.

•  Since the statement is true , shade the region contain point (0,0)

Shaded in fuchsia colour.

Now second inequality 3x - y < 2.

•  Graph the line y = 3x - 2
• Since the inequality symbol is < the boundary is not included the solution set.

Graph the boundary of the inequality 3x - y < 2 with dotted line.

• To determine which half plane to be shaded use a test point in either half- plane.

A simple choice is (0,0). Substitute x = 0 and y = 0 in original inequality

3(0) - 0 < 2

0 < 2

The statement is true.

• Since the statement is true , shade the region contain point (0,0)

Shaded in aqua colour.

Graph :

The solution of the system is the set of ordered pairs in the intersection of the graph of 2x + y ≤ 8 and 3x - y < 2. This region is shaded in light purple colour.

Answer 2

Substitution method

3) The equations are 2x + y = 7 and 3x - 2y = - 7

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

y = - 2x + 7

Find the value of x by substituting - 2x + 7 for y in the Second equation 3x - 2y = - 7.

3x - 2(- 2x + 7) = - 7

3x + 4x - 14 = - 7

7x = - 7 + 14

7x = 7

x = 1

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

First equation 2x + y = 7

2(1) + y = 7

y = 7 - 2

y = 5

Solution (x, y) = (1, 5).

So the required step is substitute - 2x + 7 for y in 3x - 2y = - 7.

Option C is correct .

Answer 3

2) Solving system of linear equations with two variables by graphing

General form of linear equation ax + by = c where a, b, c are real numbers and a and b are not both zero.

Linear function represent a line in the graph.

When we are solving system graphically need find intersection point of two lines.

Case 1 : If the lines are intersect, then the intersecting point is solution.

The system is called independent.

Example system is - x + y = 2 and - 2x + y = 1.

Solution is intersecting point (x, y) = (1, 3).

Case 2 : If the lines are parallel.

Example system - 2x + y = 2 and - 2x + y = 1

The parallel lines are does not intersect, then there is no solution.

The system is called inconsistent.

Case 3 :

Example system is 4x - 2y = 2 and 2x - y = 1

The lines are equal.They intersect at every point along their length.

This is called dependent system, there are infinitely many solutions.

Comments

Contd...

Solving system of linear inequalities by graphing

A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables.

To graph a system of inequalities fallowing steps.

Graph the each inequality separately.

Shade the regions satisfy the each inequality.

The solution of the system is the set of ordered pairs in the intersection of the graph of two inequalities.

Observe the above solution 1 for complete procedure to solving system of inequalities.

Answer 4

4)

In this case m represents number of hours plays by Macro and t represents number of hours plays by Tam.

According to given data the algebraic equations are m + t = 15 ---> (1)

m = 3 + 2t ---> (2)

Now solve the equations (1) and (2).

Substitute m = 3 + 2t in equation (1).

3 + 2t + t = 15

3 + 3t = 15

3t = 15 - 3

3t = 12

t = 12/3

t = 4

Substitute the t value in equation (1).

m + 4 = 15

m = 15 - 4

m = 11.