Word Problems with Systems: Practice

The $+y$ and $-y$ cancel when you add the equations.

$(x + y) + (x - y) = 10 + 4 \;\Rightarrow\; 2x = 14 \;\Rightarrow\; x = 7$

So $x = 7$.

"How many tickets in all" adds the counts of the two kinds.

$x + y = 50$

This is the count equation; the value equation $10x + 6y$ answers a different question.

Distance equals rate times time, so multiply the speed $b + c$ by the time $2$.

$2(b + c) = 30$

Rate times time gives distance; adding or dividing by the time does not.

The amounts of the two ingredients add up to the batch volume.

$x + y = 8$

The volumes add to $8$ liters, so $x + y = 8$.

The first equation says $y$ is the same as $x + 2$, so replace $y$ in $x + y = 8$ with $x + 2$.

$x + (x + 2) = 8$

From here $2x + 2 = 8$, so $x = 3$ and $y = 5$.

Each unknown quantity gets its own clearly defined variable, with units.

$d = \text{the number of dimes}$

Naming $d$ as the number of dimes keeps the count and value equations straight.

Multiply each count by the coin's value in cents, then add.

$5n + 10d = 95$

A nickel is $5$ cents and a dime is $10$ cents, so the value is $5n + 10d = 95$.

Break-even is the point on both lines, where the company neither loses nor gains.

$\text{cost} = \text{revenue}$

That is where the cost line and the revenue line cross.

Subtract $3x$ from both sides, then divide.

$5x = 3x + 60 \;\Rightarrow\; 2x = 60 \;\Rightarrow\; x = 30$

So break-even is at $x = 30$ items.

The sum is $15$, so $x + y = 15$. The difference, larger minus smaller, is $3$, so $x - y = 3$.

$x + y = 15, \qquad x - y = 3$

Since $x$ is the larger, the difference is $x - y$, not $y - x$.

From $2x = 18$ you get $x = 9$; back-substitute into $x + y = 15$.

$9 + y = 15 \;\Rightarrow\; y = 6$

So the pair is $(9, 6)$.

"How many in all" adds the two counts.

$a + c = 40 + 25 = 65$

Answer the actual question: $65$ tickets in all, not just one of the two counts.