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Step 11: Algebra 1

Systems of equations

Step 11 · Algebra 1 topics

Solve systems by elimination when the equations need multiplying first, set up and solve word problems with two unknowns, and graph systems of linear inequalities.

Elimination with multiples

When neither letter cancels straight away, multiply one or both equations so that the coefficients of one letter become opposites. Then add.

Example: Multiplying one equation

Solve 3x + 2y = 16 and 5x − 4y = 1.

  1. Double the first: 6x + 4y = 32.
  2. Add the second: 11x = 33, so x = 3.
  3. Then 9 + 2y = 16, so y = 3.5.

Answer(3, 3.5)

Example: Multiplying both

Solve 2x + 3y = 7 and 3x − 2y = 4.

  1. Multiply the first by 2 and the second by 3: 4x + 6y = 14 and 9x − 6y = 12.
  2. Add: 13x = 26, so x = 2.
  3. Then 4 + 3y = 7, so y = 1.

Answer(2, 1)

Word problems with two unknowns

Name the two unknowns, then write one equation for each fact in the story.

Example: A mixture

A café sells coffee for $3 and tea for $2. It sold 50 drinks for $130. How many coffees did it sell?

  1. c + t = 50 and 3c + 2t = 130.
  2. Double the first: 2c + 2t = 100.
  3. Subtract: c = 30.

Answer30 coffees, and 20 teas

Example: Two speeds

A boat goes 24 miles downstream in 2 hours and back upstream in 3 hours. Find its speed in still water and the speed of the current.

  1. Downstream: b + c = 12. Upstream: b − c = 8.
  2. Add: 2b = 20, so b = 10.
  3. Then c = 2.

AnswerBoat 10 mph, current 2 mph

Systems of inequalities

A linear inequality such as y > 2x − 1 is shown by shading one side of its line. Draw a dashed line for > or <, and a solid line for ≥ or ≤. The solution of a system of inequalities is where the shaded regions overlap.

y ≤ −x + 4 (solid line) and y > x − 2 (dashed line). The solutions are the shaded region, which includes (0, 0).

Example: Testing a point

Is (1, 2) a solution of y ≤ −x + 4 and y > x − 2?

  1. First: −1 + 4 = 3, and 2 ≤ 3 is true.
  2. Second: 1 − 2 = −1, and 2 > −1 is true.

AnswerYes

Example: Writing a system

Tickets cost $5 for students and $8 for adults. You can spend at most $40 and want at least 3 tickets. Write the system.

  1. Let s and a be the numbers of tickets.
  2. Cost: 5s + 8a ≤ 40. Count: s + a ≥ 3.

Answer5s + 8a ≤ 40 and s + a ≥ 3

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Solve 2x + 5y = 1 and 3x − 2y = 11.

    Show answer

    (3, −1)

  2. Solve 4x + 3y = 10 and 6x + 3y = 12.

    Show answer

    (1, 2)

  3. Two numbers add to 45, and one is twice the other. What are they?

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    15 and 30

  4. A plane flies 600 miles with the wind in 2 hours and back against it in 3 hours. What is the wind speed?

    Show answer

    50 mph: p + w = 300 and p − w = 200.

  5. Is (0, 0) a solution of y > 2x + 1?

    Show answer

    No: 0 > 1 is false.

  6. Is the boundary line of y < 3x − 2 solid or dashed?

    Show answer

    Dashed, because points on the line are not solutions.

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