Back to Tutor Intake Assessment: Solve real-world and mathematical problems leading to two linear equations in two variables

8.EE.C.8.c Tutor Intake -- Real-World Systems of Equations

This short check helps your tutor see where to start. Work each problem on
your own, without a calculator. If you are not sure, give your best try —
every answer helps your tutor plan your first few sessions.

Grade 8·8 problems·~9 min·Common Core Math - Grade 8·standard·8-ee-c-8c

Work through problems with immediate feedback

Concepts

A problem says: "A group bought 12 tickets total, paying $81."
Let $a$ = number of adult tickets and $c$ = number of child
tickets. A student writes only the equation $a + c = 12$ and
tries to solve for both variables. Is this enough information?

No — a single equation in two variables has many solutions (like $(1,11)$ , $(2,10)$ , etc.); a second equation (e.g., about total cost) is needed to pin down a unique answer.

Yes — since there are two variables, one equation is always sufficient to find both.

No — but only because 12 is too large a number to work with in one equation.

Yes, as long as you guess values for $a$ and $c$ that add to 12.

A student solves a ticket-sales system and finds $a = 7$ ,
$c = 5$ . They write " $a = 7$ , $c = 5$ " as their final answer to
the word problem. Is this a complete answer?

No — the answer should be interpreted in context, e.g., "7 adult tickets and 5 child tickets were sold."

Yes — once the numeric values are found, no further interpretation is needed.

No — the values themselves are incorrect and need to be recalculated.

Yes, as long as the variable letters ( $a$ , $c$ ) are shown alongside the numbers.

Procedures

A rectangle's perimeter is 46 cm. Its length is 5 cm more than
twice its width: $l = 2w + 5$ . Using the perimeter equation
$2l + 2w = 46$ , substitute for $l$ and solve for the width $w$ .

The line through $(0, 1)$ and $(2, 5)$ has slope 2. The line
through $(1, 0)$ and $(3, 4)$ also has slope 2. Do these two
lines intersect? Enter 1 for yes or 0 for no.

A student buys notebooks ($3 each) and pens ($1 each), spending
$20 total, buying 3 items in all: $n + p = 3$ and
$3n + p = 20$ . Solving gives $n = 8.5$ . Is this a valid answer
to the real-world problem? Enter 1 for yes or 0 for no.

Two friends save money: Aisha starts with $40 and saves $15 per
week ( $y = 40 + 15w$ ); Ben starts with $100 and saves $5 per
week ( $y = 100 + 5w$ ). Set the two expressions for $y$ equal and
solve for $w$ , the number of weeks until they have the same
amount saved.

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