Systems of Equations (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Systems of Equations: core idea

Elimination.

Solve $2x+y=11$ , $x-y=1$ for $x$ .

Hint: Eliminate one variable, then check both equations. Answer: 4.

1. Solve $3x+2y=16$ , $x+y=6$ for $y$ .
2. Solve $x+y=5$ , $xy=6$ for the larger possible $x$ .
3. Two numbers sum to $17$ and differ by $5$ . Find the larger.
4. Solve $4x-y=9$ , $2x+y=15$ for $x$ .
5. Solve $x+2y=13$ , $3x-y=4$ for $y$ .
6. A $2$ -dollar and a $5$ -dollar item total $19$ for $5$ items. How many $5$ -dollar items?
7. Solve $2x+3y=17$ , $x+y=7$ for $x$ .
8. Find $x$ if $x+y=12$ and $2x-y=9$ .
9. Solve $5x+2y=24$ , $x-y=3$ for $y$ .