3.2 - Solving Systems Algebraically

3.2 - Solving Systems Algebraically

Objective

To solve a linear system by substitution and elimination.

Math Response

Solve the equation $$-2x-3y=-7\$$ for y.

Type answer here

Correct Answer: y=-\frac{2}{3}x+\frac{7}{3}

Deg^°

Rad

Poll

Why is the equation $$y=-\frac{2}{3}x+\frac{7}{3}$$ difficult to graph?

• I'm bad at graphing

• If I use a table of values I will get fractions for my y values

• It has a fraction for a y-intercept

• What do you mean? It is not difficult to graph.

This is a potential problem

Last time we learned how to solve systems of equations by graphing. But there are plenty of reasons why that might nake it difficult to determine the correct solution to the system:

The trouble with graphing:

1. If your graph is not accurate, you will not have the correct point of intersection for the two lines

2. Some equations (such as those with fractions for y-intercepts) are difficult to graph just in general

3. If the coordinates of the solution are not integers (the lines cross inside one of the boxes) it is impossible to tell exactly what the coordinates are
   

So what are our options?

​Solve the system using algebra

We've got two ways we can do this:

Substitution
Elimination

Substitution

​Use this method when:

One (or both) equations are solved for one variable, or can easily be solved for one variable.

Example 1

Example 2

It can still get messy

​With fractions and the distributive property

You can see that the result of using substitution is we created an equation that only had one variable (either x or y), we solved for that and the substituted back into one of the original equations. We have a second method we can use to "eliminate" a variable.

Example 1

Example 1

Important to note

You can multiply one or both equations by anything you want to get a situation where you have opposite coefficients.

Poll

Given the system

$$2x-y=1$$

$$3x+2y=26$$

what is the most appropriate method to use to solve? Why?

• Substitution. The first equation can easily be solved for y.

• Graphing. Both equations can easily be converted to slope-intercept form.

• Elimination. Multiply the first equation by 2 and add the equations.

Poll

Select the box that most closely expresses how you feel about your ability to solve systems of linear equations algebraically:

• Bro, imma write a math textbook tonight

• Yeah, we did this when we were freshmen. I think I remember it.

• I'm gonna need some practice. It's confusing.

• No clue. None.

Deep breath....

​Use the remainder of time in class to practice any of the items on your notes pages

Then we will continue to practice in small groups next time we are together.