Systems of Linear Equations - Pre Algebra

1. Ten tickets cost 65 dollars in total: adult tickets $a$ at 8 dollars and child tickets $k$ at 5 dollars. Which system models this?

   $a + k = 10$ and $8a + 5k = 65$.

2. What check shows that $(2, 3)$ solves $x + y = 5$ and $2x - y = 1$?

   Substitution makes both true: $2 + 3 = 5$ and $2(2) - 3 = 1$.

3. Solve: $2x + y = 9$ and $2x - y = 3$.

   $(3, 3)$. Adding gives $4x = 12$, so $x = 3$ and $y = 3$.

4. How many solutions does the system $x + y = 4$ and $2x + 2y = 8$ have?

   Infinitely many. The second equation is the first multiplied by 2, so they are the same line.

5. Two numbers add to 20 and differ by 6. What are the numbers?

   13 and 7. Adding $x + y = 20$ and $x - y = 6$ gives $2x = 26$.

6. How many solutions does the system $y = 3x + 1$ and $y = -2x + 1$ have?

   Exactly one, $(0, 1)$. The slopes differ, so the lines cross once.

7. Solve by adding the equations: $x + y = 10$ and $x - y = 4$.

   $(7, 3)$. Adding gives $2x = 14$, so $x = 7$ and $y = 3$.

8. What is a solution of a system of linear equations?

   An ordered pair that makes every equation in the system true at the same time.

9. By what number should $x + 2y = 7$ be multiplied so that adding it to $3x - y = 5$ eliminates $x$?

   $-3$. This gives $-3x - 6y = -21$, and $-3x + 3x = 0$.

10. How many solutions does the system $x + y = 4$ and $x + y = 6$ have?

    None. A sum cannot equal both 4 and 6.

11. The lines $y = 3x + 2$ and $y = mx + 7$ intersect at $(1, 5)$. What is $m$?

    $m = -2$. Substitute the point: $5 = m + 7$.

12. How many solutions can a system of two linear equations have?

    Exactly one, none, or infinitely many.

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