Solving Systems by Substitution - Pre Algebra

1. In substitution, the variables cancel and leave $0 = 4$. What does this mean?

   The system has no solution; the lines are parallel.

2. Solve by substitution: $y = 3x - 1$ and $y = x + 5$.

   $(3, 8)$. Setting $3x - 1 = x + 5$ gives $x = 3$, and then $y = 8$.

3. What are the main steps of the substitution method?

   Solve one equation for one variable, substitute that expression into the other equation, solve, then substitute back to find the other variable.

4. In substitution, the variables cancel and leave $5 = 5$. What does this mean?

   The system has infinitely many solutions; the equations describe the same line.

5. Substituting $y = x + 3$ into $4x + y = 13$ gives what equation, and what is $x$?

   $4x + x + 3 = 13$, so $5x = 10$ and $x = 2$.

6. After finding the value of $x$ in substitution, how is $y$ found?

   Substitute the $x$-value into the isolated expression or either original equation and solve for $y$.

7. Solve by substitution: $y = 2x$ and $x + y = 12$.

   $(4, 8)$. Substituting gives $3x = 12$, so $x = 4$ and $y = 8$.

8. Solve by substitution: $y = x + 3$ and $2x + y = 18$.

   $(5, 8)$. Substituting gives $3x + 3 = 18$, so $x = 5$ and $y = 8$.

9. Solve $3x - y = 4$ for $y$, as a first substitution step.

   $y = 3x - 4$. Subtract $3x$ from both sides, then multiply by $-1$.

10. Solve by substitution: $x = 2y$ and $x + 3y = 20$.

    $(8, 4)$. Substituting gives $5y = 20$, so $y = 4$ and $x = 8$.

11. For $x + 4y = 9$ and $3x - 2y = 5$, which variable is easiest to isolate first?

    $x$ in the first equation, because its coefficient is 1: $x = 9 - 4y$.

Study these flashcards