ASVAB Study Guide
Algebra · Lesson 8 of 13

Solving Systems of Equations: The Substitution and Elimination Methods

Key rules and formulas, worked examples and 6 practice problems with step-by-step solutions.

1.Substitution Method

Replacing one variable with an equivalent expression from another equation.

If $$x=y+k$$ Then $$a(y+k)+by=c$$

Example

Solve the system using substitution:

$$y = 2x + 1$$

$$3x + 2y = 16$$

Show solution
  • Step 1: Substitute $$y = 2x + 1$$ into the second equation: $$3x + 2(2x + 1) = 16$$.
  • Step 2: Expand: $$3x + 4x + 2 = 16 \Rightarrow 7x + 2 = 16$$.
  • Step 3: Solve for x: $$7x = 14 \Rightarrow x = 2$$.
  • Step 4: Substitute $$x = 2$$ back into $$y = 2x + 1$$: $$y = 2(2) + 1 = 5$$.

=> Conclusion: Solution is (2, 5).

Practice problems

  1. Problem 1

    Solve using substitution:

    $$x = y + 3$$

    $$2x + y = 12$$

    Show solution

    Answer: $$(5, 2)$$

    • Step 1: Substitute $$x = y + 3$$ into $$2x + y = 12$$: $$2(y + 3) + y = 12$$.
    • Step 2: Expand: $$2y + 6 + y = 12 \Rightarrow 3y + 6 = 12$$.
    • Step 3: Solve for y: $$3y = 6 \Rightarrow y = 2$$.
    • Step 4: Calculate $$x = 2 + 3 = 5$$.

    => Conclusion: $$(x, y) = (5, 2)$$.

    => Correct Option: A ((5, 2)).

  2. Problem 2

    Solve using substitution:

    $$y = 3x - 5$$

    $$2x + 3y = 7$$

    Show solution

    Answer: $$(2, 1)$$

    • Step 1: Substitute $$y = 3x - 5$$ into $$2x + 3y = 7$$: $$2x + 3(3x - 5) = 7$$.
    • Step 2: Expand: $$2x + 9x - 15 = 7 \Rightarrow 11x - 15 = 7$$.
    • Step 3: Solve for x: $$11x = 22 \Rightarrow x = 2$$.
    • Step 4: Calculate $$y = 3(2) - 5 = 1$$.

    => Conclusion: $$(x, y) = (2, 1)$$.

    => Correct Option: A ((2, 1)).

  3. Problem 3

    Solve using substitution:

    $$x - 2y = 4$$

    $$3x + 5y = 1$$

    Show solution

    Answer: $$(2, -1)$$

    • Step 1: Isolate x in the 1st equation: $$x = 2y + 4$$.
    • Step 2: Substitute into 2nd equation: $$3(2y + 4) + 5y = 1$$.
    • Step 3: Expand: $$6y + 12 + 5y = 1 \Rightarrow 11y + 12 = 1$$.
    • Step 4: Solve for y: $$11y = -11 \Rightarrow y = -1$$.
    • Step 5: Calculate $$x = 2(-1) + 4 = 2$$.

    => Conclusion: $$(x, y) = (2, -1)$$.

    => Correct Option: A ($$(2, -1)$$)

Remember

Best used when a variable has a coefficient of 1 or -1. It 'plugs' one equation into the other to create a single-variable equation.

2.Elimination Method

Adding or subtracting equations to cancel out one variable.

$$(Ax+By)+(Cx-By)=D+E \Rightarrow (A+C)x=D+E$$

Example

Solve the system using elimination:

$$2x + 3y = 12$$

$$4x - 3y = 6$$

Show solution
  • Step 1: Add the two equations directly (since $$+3y$$ and $$-3y$$ cancel out):

$$(2x + 4x) + (3y - 3y) = 12 + 6 \Rightarrow 6x = 18$$.

  • Step 2: Solve for x: $$x = 3$$.
  • Step 3: Substitute $$x = 3$$ into $$2x + 3y = 12$$: $$2(3) + 3y = 12 \Rightarrow 6 + 3y = 12$$.
  • Step 4: Solve for y: $$3y = 6 \Rightarrow y = 2$$.

=> Conclusion: Solution is (3, 2).

Practice problems

  1. Problem 4

    Solve using elimination:

    $$3x + 2y = 11$$

    $$5x - 2y = 13$$

    Show solution

    Answer: $$(3, 1)$$

    • Step 1: Add the two equations to eliminate y: $$(3x + 5x) = 11 + 13 \Rightarrow 8x = 24$$.
    • Step 2: Solve for x: $$x = 3$$.
    • Step 3: Substitute $$x = 3$$ into $$3(3) + 2y = 11$$: $$9 + 2y = 11 \Rightarrow 2y = 2 \Rightarrow y = 1$$.

    => Conclusion: $$(x, y) = (3, 1)$$.

    => Correct Option: A ((3, 1)).

  2. Problem 5

    Solve using elimination:

    $$2x + 3y = 12$$

    $$2x - y = 4$$

    Show solution

    Answer: $$(3, 2)$$

    • Step 1: Subtract the 2nd equation from the 1st to eliminate x: $$(3y - (-y)) = 12 - 4 \Rightarrow 4y = 8$$.
    • Step 2: Solve for y: $$y = 2$$.
    • Step 3: Substitute $$y = 2$$ into $$2x - 2 = 4$$: $$2x = 6 \Rightarrow x = 3$$.

    => Conclusion: $$(x, y) = (3, 2)$$.

    => Correct Option: A ((3, 2)).

  3. Problem 6

    Solve using elimination:

    $$3x + 4y = 18$$

    $$2x - 3y = -5$$

    Show solution

    Answer: $$(2, 3)$$

    • Step 1: Multiply 1st eq by 3 and 2nd eq by 4 to eliminate y:

    $$9x + 12y = 54$$

    $$8x - 12y = -20$$

    • Step 2: Add the equations: $$17x = 34 \Rightarrow x = 2$$.
    • Step 3: Substitute $$x = 2$$ into $$3(2) + 4y = 18 \Rightarrow 6 + 4y = 18 \Rightarrow 4y = 12 \Rightarrow y = 3$$.

    => Conclusion: $$(x, y) = (2, 3)$$.

    => Correct Option: B ((2, 3)).

Remember

Best used when equations are in Standard Form (Ax + By = C). Look for matching coefficients with opposite signs to 'eliminate' a variable.