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Systems OF EQUATIONS. SOLVING SYSTEMS BY GRAPHING. WARM UP. 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?

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Systems of equations

Systems OF EQUATIONS

SOLVING SYSTEMS BY GRAPHING


WARM UP

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


WHAT IS A SYSTEM?

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.


  • NUMBER OF SOLUTIONS

  • 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.


Example one: one solution consistent. A consistent system can be either independent or dependent.

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


Got it? consistent. A consistent system can be either independent or dependent.

Graph: 2x – y = -5 and -2x – y = -1

Check your solution!


Example two: infinitely many solutions consistent. A consistent system can be either independent or dependent.

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.


Got it? consistent. A consistent system can be either independent or dependent.

What is the solution of the system.

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

Describe the number of solutions.


Example three: no solutions consistent. A consistent system can be either independent or dependent.

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.


Got it? consistent. A consistent system can be either independent or dependent.

What is the solution of the system.

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

Describe the number of solutions.


Time to write. consistent. A consistent system can be either independent or dependent.

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


Ticket out the door: consistent. A consistent system can be either independent or dependent.

Complete the worksheet Solving Systems of Linear Equations.


Homework: consistent. A consistent system can be either independent or dependent.

Complete the worksheet – Solving Systems of Equations by Graphing.


CONCEPT SUMMARY: SYSTMES OF LINEAR EQUATIONS consistent. A consistent system can be either independent or dependent.


WARM UP consistent. A consistent system can be either independent or dependent.

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


WARM UP solution, no solutions or infinitely many solutions. Solve for y and then graph.

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 solution, no solutions or infinitely many solutions. Solve for y and then graph.

  • 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.


Answer these questions….. 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?



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