Pre-class Work

Read and work through checkpoint exercises at the following link

Section 8.1: Systems of Linear Equations in Two Variables

When it says to use your calculator to graph, you may use the graphing calculator at https://www.desmos.com/

4.3 Examples

Example 4.3.1

A farmer has pigs and chickens. If she counts 14 heads and 40 legs total, how many of each animal does she have?  Solve this problem in two ways as noted below. Share your solutions in your group and discuss how they compare/contrast.

a) Solve without using any variables and in a way that an elementary school student could understand. Draw pictures and explain your calculations. Clearly document any trial and error (guessing and checking) involved.

b) Set up two equations, with two variables, to represent the given information. Clearly state exactly what each variable represents. Then solve for the variables using the substitution or elimination method. Show your work below. If you need a refresher on these methods, refer to the pre-class work or the review after Example 4.3.2.

c) Solve the problem by graphing the corresponding equations. Sketch your graph and justify your answer below.

Example 4.3.2

Sandrine “accidentally” broke her piggy bank to find a combined total of 42 dimes and quarters. If the coins totaled $8.25, how many dimes and how many quarters did she have in her piggy bank?

a) Solve without using any variables and in a way that an elementary school student could understand. Draw pictures and explain your calculations. Clearly document any trial and error (guessing and checking) involved.

b) Set up two equations, with two variables, to represent the given information. Clearly state exactly what each variable represents. Then solve for the variables using the substitution or elimination method.  Show your work below. If you need a refresher on these methods, refer to the pre-class work or the review after this example.

c) Solve the problem by graphing the corresponding equations. Sketch your graph and justify your answer below.

Review of Linear Systems (with two equations)

A 2 by 2 linear system (of equations) is a set of 2 linear equations with the same 2 variables.

Illustration:

[latex]x + 2y = 12[/latex]

[latex]2x - 4y = 3[/latex]

A solution of a 2 by 2 linear system is an ordered pair of values for the variables that makes each equation in the system true. As shown in the pre-class work, three ways to solve such systems are…

1. Using the substitution method
2. Using the elimination method
3. Graphing

Below we review these three methods

Substitution Method Review/Refresher

Khan Academy Video (~10 min):

Summary of steps:

1. Solve one of the equations for one of the variables in terms of the other.
2. Substitute the expression resulting from step (1) into the second equation; doing so yields an equation with one variable only.
3. Solve the new equation.
4. Plug your result from step (3) into the equation from step (1) to find the other variable.

Example 4.3.3

Use the substitution method to solve the following systems of equations.

a)

[latex]4x+3y=96[/latex]

[latex]x+y=27[/latex]

b)

[latex]2x-y=2[/latex]

[latex]4x-4y=-22[/latex]

The method of substitution is convenient if one of the variables in the system has a coefficient of 1 or -1, because it is easy to solve for that variable. If none of the coefficients is 1 or -1, then the elimination method is usually more efficient.

 Elimination Method Review/Refresher

Khan Academy Video (~12 min):

Summary of the Steps:

1. Choose one of the variables to ‘eliminate’. Multiply one or both equations by a suitable factor so that the coefficients of that variable are opposites in the equations. The factor for each equation may be different.
2. Add the two new equations together which will eliminate the variable chosen in step 1.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value found in step 3 into either of the original equations and solve for the other variable.

Example 4.3.4

Use the elimination method to solve the following systems of equations.

a) [latex]5x-2y=-4[/latex]

[latex]-6x+3y=5[/latex]

b) [latex]3x-4y=-11[/latex]

[latex]2x+6y=-3[/latex]

Graphing Method Review/Refresher

Khan Academy Video (~12 min):

Summary of the Steps:

1. Graph the lines given by each equation on the same set of axes.
2. Find the intersection point(s) of the two lines, if they exist. This gives you your solution(s)

Example 4.3.5

Check your answers from Example 4.3.3 and 4.3.4 by graphing in Desmos.  Record any discrepancies below as points for discussion.

In the examples above, each system has exactly one solution. In the examples below, we explore other scenarios.

Example 4.3.6

Solve each system below using substitution or elimination and then by graphing.  Explain what you see.

a) [latex]x + y = 5[/latex]

[latex]y = 4x[/latex]

b) [latex]2x + 4y = 0[/latex]

[latex]x + 2y = 0[/latex]

c) [latex]x = y[/latex]

[latex]x - y = 6[/latex]

Example 4.3.7

Graphically explain (in general) how a 2 by 2 linear system could have each of the following scenarios and then discuss how you can tell which scenario you are in by looking at the equations.

Exactly one solution:

No solutions:

An infinite number of solutions:

One may also solve such problems using numerical methods, logic and possibly some trial and error (guess and check) as we did in Examples 4.3.1 and 4.3.2.

The ability to solve linear systems in this manner depends on the level of complexity.  For example, if the solution involves decimals or fractions that are not whole numbers, then it may be difficult, if not impossible to solve the system by trial and error.  However, if the answers are relatively small whole numbers, it is feasible to solve such problems without knowing anything about algebra or graphing.  Hence, such problems may be introduced in the elementary school classroom.

Let us look at another application at this level.

Example 4.3.8

Pablo paid $33 for 3 chocolate bars and 9 cookies. Brenda paid $48 for 12 chocolate bars and 8 cookies. Find the cost of one chocolate bar and the cost of one cookie.  Assume each costs a whole dollar amount (no cents).  Solve this problem in two ways as noted below.

a) Solve without using any variables in a way that an elementary school student could understand. Draw pictures and explain your calculations. Clearly document any trial and error (guessing and checking) involved.

b) Solve using the substitution or elimination method. Clearly state exactly what each variable represents.