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5.1 – Two-Variable Systems: Graphing

Key Terms

  • Axes – The reference lines used in a graph. (Axes is the plural of axis.)
    • The axes of a coordinate graph are perpendicular.
    • The x-axis is horizontal, and the y-axis is vertical.
  • Intersect – To cross over one another.
  • System of Two Linear Equations – Groups of two linear equations that have the same variables and are used together to solve a problem.
    • A linear equation can be written in the form y = mx + b.
    • The graph of a linear equation is a straight line.

 

Review

  • Coordinates are alphabetical.  x comes first, then y: (x, y)
    • On a graph, you plot x first (horizontally), then y (vertically).
    • You mark this point with a dot and name the coordinates with their respective numbers for (x, y).
    • Ex.  (2, -3) is located on the graph below.

Alg1B - 5.1 Solution Set Ex2

 

Notes

  • Solutions Sets
    • One Solution – when multiple lines cross at one single point
      • They have unique slopes and y-intercepts.

Alg1B - 5.1 Solutions One

    • Zero Solutions – when multiple lines never cross (they are parallel)
      • They have the same slope, but different y-intercepts.

Alg1B - 5.1 Solutions Zero

    • Infinitely Many Solutions – when multiple lines lie on top of one another and share all of the same points
      • They have the same equation; so, same slope and same y-intercept.

Alg1B - 5.1 Solutions Infinitely Many

 


Alg1B - 5.1 Slope Graphing

 


Examples

  • Ex 1. What is the solution to the system of equations graphed below?
    • Line 1:  y = 2x – 7
    • Line 2:  y = -x – 1
    • It’s easy to see the answer by tracing the x and y coordinate to the point where the lines meet.  They meet at (2, -3).
    • However, there is another way to solve for the solution point if you can’t trace it on a graph:
      • Since y is equal to both “2x – 7” and also “-x – 1,” you can set both equations equal to one another:
        • Setup: 2x – 7 = -x – 1
        • Add 7 to both sides: 2x = -x + 6
        • Add x to both sides: 3x = 6
        • Divide both sides by 3: x = 2
        • Substitute x = 2 into either of the two equations.  Let’s pick y = 2x – 7.
          • y = 2 (2) – 7
        • Simplify: y = 4 – 7; so, y = -3
        • So, x = 2 and y = -3
          • Answer in coordinate form: (2, -3)

Alg1B - 5.1 Solution Set Ex

 

  • Ex 2. The owner of a bike shop sells unicycles and bicycles and keeps inventory by counting seats and wheels.
    • One day, she counts 20 seats and 28 wheels.
    • a. Which equation in slope-intercept form shows that the number of bicycles (x) plus the number of unicycles (y) is 20?
      • Setup: x: bicycles and y: unicycles
      • The total is 20, so x + y = 20.
      • Rearrange to slope-intercept form by subtracting x on both sides.
      • Answer: y = -x + 20
    • b. Which equation in slope-intercept form shows that x bicycles and y unicycles together have a total of 28 wheels?
      • Setup: x: bicycles and y: unicycles
      • 2x would represent the wheels on a bicycle
      • y would represent the wheels on a unicycle
      • The total wheel count is 28; so, 2x + y = 28.
      • Rearrange to slope-intercept form by subtracting 2x on both sides.
      • Answer: y = -2x + 28

 


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