Systems - GED Math

1. Solve the system $x + y = 10$ and $x - y = 4$.

   $x = 7$, $y = 3$. Adding the equations gives $2x = 14$.

2. How do you check a proposed solution to a system?

   Substitute the $x$ and $y$ values into both equations. It is a solution only if both are true.

3. What are the steps of the substitution method?

   Solve one equation for one variable, substitute that expression into the other equation, and solve.

4. What is the solution of a system of two linear equations?

   The ordered pair $(x, y)$ that makes both equations true. On a graph, it is where the lines intersect.

5. What is $x$ in the system $2x + 3y = 12$ and $4x - 3y = 6$?

   $x = 3$. Adding the equations cancels $y$: $6x = 18$.

6. Solve the system $y = 2x$ and $x + y = 12$.

   $x = 4$, $y = 8$. Substitute: $x + 2x = 12$, so $3x = 12$.

7. What is the goal of the elimination method?

   To add or subtract the equations (after multiplying if needed) so that one variable cancels out.

8. How many solutions does a system have if its lines have the same slope but different $y$-intercepts?

   None. The lines are parallel and never intersect.

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

   Infinitely many. The second equation is the first multiplied by $2$, so both describe the same line.

10. Adult tickets cost 10 dollars and child tickets 6 dollars. Sales were $50$ tickets for 380 dollars. Write a system using $a$ and $c$.

    $a + c = 50$ and $10a + 6c = 380$. One equation counts tickets; the other counts dollars.

11. Solve the system $y = x + 1$ and $y = -x + 5$.

    $x = 2$, $y = 3$. Set the right sides equal: $x + 1 = -x + 5$, so $2x = 4$.

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