Linear Systems Lesson 2: Solving Systems of Linear Equations Graphically

Today’s Objectives:

Students will be able to solve problems that involve systems of linear equations in two variables graphically and algebraically, including:

·  Determine and verify the solution of a system of linear equations graphically, with and without technology

·  Explain the meaning of the point of intersection of a system of linear equations

Vocabulary: system of linear equations, linear system

Solving Systems of Linear Equations Graphically

The solution of a linear system can be estimated by ______ both equations on the same ______. If the two lines intersect, the coordinates (x,y) of the point of intersection are the ______ of the linear system.

Each equation of the following linear system is graphed on the grid.

3x+2y=-12 (1)

-2x+y=1 (2)

We can use the graphs to ______ the solution of the linear system.

The set of points that satisfy both equations lie where the two graphs ______. From the graph, the point of intersection appears to be ______. To verify the solution, we check that the coordinates ______satisfy both equations:

For each equation, the left side equals the right side. Since x = ___ and y = ___ satisfy each equation these numbers are the solution of the linear system.

Example 1: (You do) Solving a Linear system by Graphing

Solve the linear system

x+y=8 (1)

3x-2y=14 (2)

Solution:

Example 2: Solving a Problem by Graphing a Linear System

One plane left Shanghai at noon to travel 1400 km to Urumqi at an average speed of 400 km/h. Another plane left Urumqi at the same time to travel to Shanghai at an average speed of 350 km/h. A linear system that models this situation is:

where d is the ______ in km’s from Shanghai to Urumqi and t is the ______ in hours since the planes took off.

a)  Graph the linear system above.

b)  Use the graph to solve this problem: When do the planes pass each other and how far are they from Urumqi?

Solution:

The planes pass each other when they have been travelling for the ______ time and they are the same distance from Urumqi.

a)  Solve the linear system to determine values of d and t that satisfy both equations.

For the graph of equation (1), the slope is ______ and the vertical intercept is ______.

For the graph of equation (2), the slope is ______ and the vertical intercept is _____.

b)  The graphs appear to intersect at ______; that is, the planes appear to pass each other after travelling for ____ hours and at a distance of ____ km from Urumqi. Use the coordinates of the point of intersection to verify the solution.

These times and distances are ______ because these measures cannot be read ______ from the graph.

The planes pass each other after travelling for approximately ______ and when they are approximately ______km from Urumqi.

a)  Write a linear system to model this situation:

To visit the Yu Garden in Shanghai, the ticket price is \$5 for a student and \$9 for an adult. In one hour, 32 people entered the garden and a total of \$180 in admission fees was collected.

b)  Graph the linear system then solve this problem:

How many students and how many adults visited the garden during this time?

Solution:

a)  The linear system is:

b)  Use intercepts to graph each line:

Equation / a-intercept / s-intercept

The total number of people is ____and the total cost is ______, so the solution is correct. ____ students and ___ adults visited the garden.

Homework: pg. 409-410, # 5-17

Next class: xiao quiz