** Key Concepts**

- Solve a System by Graphing
- Graph a system of equations with the unique solution
- Graph a system of equations with no solution
- Graph a system of equations with Infinitely Many Solutions

**Solving System of Linear Equations by Graphing**

- A ……………….. is formed by two or more linear equations that use the same variables.

- A ………… of a system of linear equations is any ordered pair that makes all equations in the system

- The lines intersect at one point. Then the system has ………. solution.

- The lines do not intersect; they are parallel. This system has………… solution.

- The lines intersect at every point; they are the same line. This system has infinitely …….. solutions.

- Slope and intercept form of –2x + y=5 is ……………

- Compare with slope and intercept form for solutions

- How many solutions for y=2x+4 and y=3x-1…………..

- How many solutions for y=3x+4 and y=3x+5………….

- How many solutions for y=3x+4 and y=4+3x…………

- Graph the system of equations to determine the solution. x+4y=8 and 3x+4y=0

**Answers:**

- System of linear equations

- Solution

- One solution

- No solution

- Many solutions

- y=2x+5

- One solution

- No solution

- Infinitely solutions

- I don’t know. Ok, let’s draw to find solutions.

**Explore It! **

Beth and Dante pass by the library as they walk home using separate straight paths.

Model with Math

The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library.

**Solution:**

B. Write a system of equations that represents the paths taken by Beth and Dante.

System of equations

- -x-2y=-8 and 5x-2y=4

- -x-y=-5 and x-y=-1

Beth and Dante can choose infinite paths to choose to go from library to home.

Focus on math practices reasoning. What does the point of intersection of the lines represent in the situation?

The point of intersection of their paths is called the solution for the equations.

In this lesson, we can draw graphs for finding solutions for the system of equations.

Systems of equations: A set of equations is called a system of equations. The solutions must satisfy each equation in the system.

Systems of linear equations:

A solution to a system of equations is an ordered pair that satisfies all the equations in the system.

A system of linear equations can have:

- Exactly one solution

- No solutions

- Infinitely many solutions

Systems of linear equations:

There are four ways to solve systems of linear equations:

- By slope and intercept form

- By graphing

- By substitution

- By elimination

Graphical method for solving linear equations in two variables:

The steps to solve linear equations in two variables graphically are given below:

**Step 1: **

To solve a system of two equations in two variables graphically, we graph each equation.

**Step 2:**

To graph an equation manually, first convert it to the form y= mx+b by solving the equation for y.

**Step 3: **

Start putting the values of *x* as 0, 1, 2, and so on and find the corresponding values of y, or vice-versa. I.e., find ordered pairs that satisfy each of the equations.

**Step 4:**** **

Plot the ordered pairs and sketch the graphs of both equations on the same axis. Identify the point where both lines meet.

**Step 5: **

The coordinates of the point or points of intersection of the graphs are the solution or solutions to the system of equations.

Essential question: How does the graph of a system of linear equations represent its solution?

**Example 1:**

Li is choosing a new cell phone plan. How can Li use the graphs of a system of linear equations to determine when the phone plans cost the same? Which plan should Li choose? Explain.

**Solution:**

Model with Math: You can use the graphs of a system of linear equations to compare the costs of each plan.

**Step 1**

Write a system of equations. Let x = the number of minutes used each month.

Let y = the total monthly cost of the plan.

Total monthly cost of plan = Cost of cost for minutes + unlimited data used

The system of equations is y = 0.20x+75

y = 0.25x+70

**Step 2:** Find the co-ordinates for the equations

- y = 0.20x+75

If x=-20, then y=0.20 X (-20) +75 =-4+75=71. Find co-ordinates by trial and error method.

**Step 3:**

- y = 0.25x+70

If x=-40, then y=0.25 X 60 +70 =-40 +70 = 60 ….. Similarly, we can find coordinates.

Graph the system.

The point of intersection, (100, 95), is the solution of the system.

The lines intersect, so there is one solution for this system of equations.

Solve a system by graphing

Graph the system.

The point of intersection, (100, 95), is the solution of the system.

The lines intersect, so there is one solution for this system of equations.

If Li uses 100 minutes each month, both plans would cost $95. She could choose either plan.

If Li uses fewer than 100 minutes, she should choose Company B. If Li uses more than 100 minutes, she should choose Company A.

**Example 2**

Solve the system. y = x – 4 and 2x – 2y = – 2. Graph the equations of the system to determine the solution.

**Solution:**

**Step 1:** Write slope – intercept form of given equations

y =x – 4.

Reasoning: What can you say about the slopes and y-intercepts of the equations?

**Step 2:** First write slope-intercept form of the given equation.

2x – 2y = – 2

x – y = -1

– y= – 1 – x

y = x + 1

If x=0, then y = 0 + 1 = 1

y = x + 1

**Graph a system of equation with no solution **

**Step 3:**

Draw by using coordinates of system of equations.

The lines are parallel. There is no point of intersection.

I.e., the lines do not intersect; they are parallel.

This system has no solution.

**Example 3**

Solve the system. y =

1212

x+3 – 3x + 6y = 18. Graph the equations of the system to determine the solution.

**Solution:**

**Step 1:** This equation is in the form of slope-intercept form.

**Step 2:**

Each (x, y) point on the line represents a solution.

The lines are the same, so the system has infinitely many solutions.

Let’s check your knowledge

- Solve each system by graphing. Describe the solutions.

5x+y=-3 and 10x+2y=-6

- Solve the system by graphing

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

- Solve the system by graphing

y = x + 5

y = x + 10

**Answers:**

- Solve each system by graphing. Describe the solutions. 5x+y=-3 and 10x+2y=-6

**Solution:**

**Step 1:**

5x+y=-3

y=-5x-3

**Step 2:**

10x+2y=-6

y=

−10x−62−10x−62

or y = -5x – 3

**Step 3: **Draw the graph for given equations.

The lines are the same, so the system has infinitely many solutions.

- Solve the system by graphing y = 3x + 5 and y=2x + 4

**Solution:** Step 1

y = 3x + 5

y=2x + 4

The lines intersect, so there is one solution for this system of equations.

Solution x=-1 and y=2

- Solve the system by graphing

y = x + 3

y = x + 10

**Solution:**

**Step 1**

y = x + 3

**Step 2 **

y = x + 10

**Step 3: **

Plotting the points on the graph

In the graph, the lines are not intersecting. So, the given system of equations doesn’t have a solution.

### Key concept

The solution of linear equation is the point of intersection of the lines defined by the equations.

**Key concepts covered**

- Solve a system by graphing
- Graph a system of equations with a unique solution
- Graph a system of equations with no solution
- Graph a system of equations with infinitely many solutions

## Exercise:

- Solve each system by graphing. Describe the solutions.

- x +y=10
- x – y= 4

- 5x+7y=50
- 7x +5y = 46

- Solve the system by graphing. Describe the solutions.

9x + 3y + 12 = 0 and 18x + 6y + 24 = 0.

- Solve the system by graphing. Describe the solutions.

3x + 2y = 5 and 2x – 3y = 7

- Solve the system by graphing. Describe the solutions.

2x – 3y = 8 and 4x – 6y = 9

- Find the solution of -x-3y=-12 and 5x-3y=6 by using the graph below.

- Find the solution of -2x-5y=-20 and 4x-5y=10 by using the graph below.

- Solve the system by graphing. Describe the solutions.
- y = -2x – 2 and y =x – 5
- The pair of linear equations px+2y=5 and 3x+y=1 has unique solution if

A) p≠6 B) p=6 C) p=5 D) p≠5

- If x=1, then the value of
*y*satisfying the equation 5/x+ 3/y =6

A) 3 B) 1/3 C)-1/3 D) 1

**Concept map:**

**What Have We Learned**

- Understand facts
- Understand how to solve a system by graphing
- Understand how to graph a system of equations with a unique solution
- Understand how to graph a system of equations with no solution
- Understand how to graph a system of equations with infinitely many solutions

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