Algebra
Systems of equations and inequalities
Writing systems of equations
20 practice questions
2 video lessons
Theory + worked examples
Theory
A system of equations is two or more equations sharing the same variables. To write one:
- Define a variable for each unknown.
- Write one equation for each condition in the problem.
- The solution satisfies all equations.
Each sentence usually becomes one equation.
Steps to write a system.
A worked setup.
A two-equation system:
\[\begin{cases}a_1x+b_1y=c_1\\a_2x+b_2y=c_2\end{cases}\]
The solution must make every equation true.
How to write a system
- Identify the unknowns and name them.
- Turn each condition into an equation.
- Check every condition is represented.
- Solve the system to answer the problem.
Example 1 — Two conditions
Adult \(a\) and child \(k\) tickets: \(2\) adults and \(3\) kids cost \(\$34\); \(1\) adult and \(1\) kid cost \(\$13\). Write the system.
Solution
One equation per sentence.
| \(2a+3k\) | \(=\) | \(34\) |
| \(a+k\) | \(=\) | \(13\) |
Example 2 — Sum and difference
Two numbers sum to \(20\) and differ by \(4\). Write the system.
Solution
Translate each statement.
| \(x+y\) | \(=\) | \(20\) |
| \(x-y\) | \(=\) | \(4\) |
Example 3 — Define variables
Why define variables first?
Solution
So each equation clearly represents a condition of the problem.
Example 4 — A mixture
You buy \(x\) pens at \(\$2\) and \(y\) notebooks at \(\$3\), spending \(\$17\) on \(7\) items. Write the system.
Solution
Cost equation and count equation.
| \(2x+3y\) | \(=\) | \(17\) |
| \(x+y\) | \(=\) | \(7\) |
Common pitfalls
Define variables clearly before writing equations.
One equation per condition — don't merge them.
The solution must satisfy all equations.
Frequently asked questions
What is a system of equations?
Two or more equations sharing the same variables.
How many equations do you need?
Usually one per unknown.
What should you do first?
Define a variable for each unknown.
What does the solution satisfy?
Every equation in the system.
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