Write systems of equations from contexts

A learner translates a contextual situation into a system of two linear equations by defining variables, identifying the quantities and relationships involved, and expressing shared constraints in forms such as $ax+by=c$ or $y=mx+b$ , with attention to units and the meaning of constants and coefficients. The learner understands that both equations model the same situation and that their simultaneous solution represents values satisfying all stated conditions; the scope excludes nonlinear systems, three or more variables, and parameterized or abstract systems.

Detailed Explanation: Write systems of equations from contexts

To write a system of equations from a context:

Define the variables and include what each variable represents.
Find two different relationships in the situation.
Translate each relationship into an equation.
Check the units and make sure both equations describe the same situation.

Example

A school play sells adult tickets for $8 and student tickets for $5. A total of 120 tickets are sold for $810. How can you write a system of equations?

Step 1: Define the variables

Let

$x=$ number of adult tickets sold
$y=$ number of student tickets sold

Step 2: Write an equation for the total number of tickets

The number of adult tickets plus the number of student tickets is 120:

x+y=120

Both $x$ and $y$ count tickets, so the result is also a number of tickets.

Step 3: Write an equation for the total money

Each adult ticket brings in $8, so adult tickets bring in $8x$ dollars.

Each student ticket brings in $5, so student tickets bring in $5y$ dollars.

The total is $810:

8x+5y=810

The coefficients $8$ and $5$ are prices per ticket, and the constant $810$ is the total amount of money.

Step 4: Write the system

The situation is represented by the system

\begin{cases} x+y=120\\ 8x+5y=810 \end{cases}

Both equations must be true at the same time. Their solution gives the numbers of adult and student tickets that satisfy both the ticket total and the money total.

Learn by doing: Write systems of equations from contexts

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Algebra Meals - 3 Meals, 2 Items (Multiple, Complex Substitution, Compound Answer), to Equation

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