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### Systems OF EQUATIONS

SOLVING SYSTEMS BY GRAPHING

Solve for y: 8x + y = -4

What do you notice about these two lines: y = -3x + 4 and

y = -3x -2?

3. Solve for y: 2y – x = 2 and y = x + 1. What do you notice about theses two lines?

4. Graph the line y = -2x + 3

Two or more equations with the same set of variables are called a system of equations. For example, y = 4x and y = 4x + 2 together are a system of equations.

Any ordered pair that makes all of the equations true is a solution of a system of linear equations.

You can estimate the solution of a system of equations by graphing the equations on the same co-ordinate plane. The ordered pair for the point of intersection of the graphs is the solution of the system because the point of intersection simultaneously satisfies both equation.

- The graph of a system of equations indicates the number of solutions.
- If the lines intersect, there is one solution.
- If the lines are parallel, there is no solution.
- If the lines are the same, there are an infinite number of solutions.

A system of equations that has at least one solution is consistent. A consistent system can be either independent or dependent.

A consistent system that is independent has exactly one solution.

A consistent system that is dependent has infinitely many solutions.

A system of equations that has no solution is inconsistent.

What is the solution of the system? Use a graph. y = x + 2 and y = 3x – 2

Graph both equations in the same co-ordinate plane.

Find the point of intersection. The lines appear to intersect at (2,4). Check to see if (2,4) makes both equations true.

y = x + 2 y = 3x – 2 (2,4) is a solution to the system.

4 = 2 + 2 4 = 3(2) - 2

4 = 4 4 = 6 - 2

4 = 4

Example two: infinitely many solutions

What is the solution to the system? Use a graph. 2y – x = 2 and y = x + 1

The equations represent the same line. Any point on the line is a solution of the system. So, there are infinitely many solutions. The system is consistent and dependent.

What is the solution of the system.

Use a graph. y = 3x – 3 and 3y = 9x - 9

Describe the number of solutions.

Graph the equations y = 2x + 2 and y = 2x -1 on the same co-ordinate plane. What is the solution to the system?

The graph of the equations y = 2x + 2 and y = 2x – 2 are parallel lines, so there is no solution. The system is inconsistent.

What is the solution of the system.

Use a graph. y = -x – 3 and y = -x + 5

Describe the number of solutions.

Before graphing the equations, how can you determine whether a system of equations has exactly one solution, infinitely many solutions, or not solutions?

Complete the worksheet Solving Systems of Linear Equations.

Complete the worksheet – Solving Systems of Equations by Graphing.

Solve for y: 2x – y = 3. Graph your line.

Yesterday you learned about systems of equations, explain what a systems is.

Explain the “number of solutions” a system could have. By just looking at the equations, how can you decide how many solutions a system has?

Simplify:

Simplify: 4(2r – 8) = (49r + 70)

Let’s practice converting equations. Solve the following equations for y.

6x + y = 5

2y = 6x – 2

9 – 3y = 3x

-2y = 4x + 5

3x + y = 8

Now, decide if each of the following systems has one solution, no solutions or infinitely many solutions. Solve for y and then graph.

3x – 2y = 6 and x + y = 2

2x – y = 1 and 4x – 2y = 2

y = 2x + 1 and y = 2x – 3

y = -4x + 5 and y = 3x – 9

x + y = 4 and 2x + 2y = 10

Does the following system have one solution, no solutions or infinitely many solutions? y = x – 1 and y = -x + 1

Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. 2x – y = 1 and y = -3

Solve: 6 – 3x = 21

Student Council is selling T-shirts during Spirit Week. It costs $20 for the design and $5 to print each shirt. Write and graph an equation.

GRAPHING USING A GRAPHING CALCULATOR

- To use a graphing calculator to find a point of intersection of two lines follow the steps below. (2ndMem – 7 – 2 – 1)
- Select Y= on your calculator.
- Select GRAPH on your calculator.
- Enter your functions into Y1 and Y2 and then press GRAPH. (If your equation is not in the form y = mx + b then you must solve for y first and then enter your equations.)
- Select 2nd – CALCULATE – INTERSECT.
- Select 1st curve? – enter.
- Select 2nd curve? – enter.
- Select Guess? – enter.
- Write down the point that is given.

Use a graphing calculator to graph each set of equations in the table. Find the point of intersection of the two lines.

1. Explain what the point of intersection represents.

How can I use a graphing calculator to find one solution for a set of two equations?

How can you use a graph to solve a system of equations?

Graph the following systems and then use your calculator to check your point of intersection.

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