A context is represented by two linear equations whose variables denote unknown quantities, with coefficients, constants, and units reflecting the relationships described, such as totals, rates, costs, differences, or comparisons. The learner understands that the system models two conditions simultaneously and that its solution is the pair of values satisfying both equations, while distinguishing equations from isolated expressions and preserving the meaning of each quantity. The scope excludes nonlinear, three-variable, parametric, and other advanced systems.
To write a system of linear equations from a context:
A school play sold 40 tickets. Adult tickets cost $8 each, and student tickets cost $5 each. The school collected $260. How can you write a system of equations?
Step 1: Define the variables.
Let
Step 2: Use the total number of tickets.
The total number of tickets is 40, so:
This equation represents the number condition.
Step 3: Use the total money collected.
Adult tickets bring in dollars, and student tickets bring in dollars. Together, they bring in $260:
This equation represents the money condition.
Step 4: Write the system together.
The two equations must be true at the same time. A solution is a pair that satisfies both conditions. For example, and works because:
and
So, always define the variables first, then translate each separate condition into an equation. An expression such as shows a calculation, but it is not an equation until it is set equal to a value.
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