# 1.3. Precalculus: Systems of linear equations

### Learning Objectives

You should be able to

• identify when an equation is linear
• express the equation of a line in multiple ways and draw their graphs
• understand when a system of two linear equations in two unknowns might have none, one or infinitely many solutions
• solve systems of simultaneous linear equations by substitution and by Gaussian elimination

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.
Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 1.3.1. Linear relationships and their graphs

Identify linear equations such as $$ax=k$$ or $$ax+by=k$$.
Describe graphs of linear equations in two variables in the forms $$y=mx+c$$, $$y-y_0=m(x-x_0)$$ and $$ax+by=c$$.
See graphically why a system of two linear equations in 2 variables may have no solutions, one unique solution of infinitely many solutions.

#### 1.3.2. Solving linear systems by substitution

Solve systems of linear equations by substituting expressions for one variable in terms of the other variables to eliminate it from the other equations.

#### 1.3.3. Solving linear systems by Gaussian elimination

Manipulate a system of linear equations to eliminate each variable from as many equations as possible

• Multiply an equation by a nonzero constant
• Add or subtract a multiple of one equation from another

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to. The additional resources gives links to external sources of information and some of these may also have further exercises to try.