Linear Equations: Forms, Graphs, and Systems

Posted on Feb 12, 2025 in Mathematics

This was made by Shawn

Equation of a Line in Slope Y-Intercept Form

Recall: Slope y-intercept form

Write the equation of the line.

a) m = -5, b = 2 => y = -5x + 2

b) Y-int (b) = -3, slope (m) = 1/2

y = (1/2)x – 3

c) Rate of change (m) = -2/5, initial value (b) = 0

y = (-2/5)x + 0 => y = -2/5x

y = 5/7x + 2

Equation of a Line in Standard Form

To graph a linear equation in standard form, you must rewrite it into slope y-intercept form.

Steps:

  1. Isolate the term with the dependent variable on one side of the equation.
  2. Divide every term in the entire equation by the coefficient of the dependent term.
  3. Separate the coefficient from the independent variable.

Parallel and Perpendicular Lines

Parallel Lines

Parallel lines are straight lines that run in the same direction; they will NEVER cross/intersect each other.

Parallel lines have the same slope value (m); slopes are equal.

Ex. Are these lines parallel?

a) y = –2x + 1, y = 3x + 1 => Not parallel

b) y = 4x – 1, y = –4x + 5 => Not parallel

c) y = 7x – 4, y = 7x + 10 => Parallel lines

Perpendicular Lines

Ex. What is the negative reciprocal of:

a) m = 1/2 => m⟈ = -2/1 => m⟈ = -2

b) m = -3/4 => m⟈ = 4/3

c) m = 5/1 => m⟈ = -1/5

Ex. Determine if the set of lines are parallel, perpendicular, or neither.

a) y = 3x – 1, y = (1/3)x + 1 => Neither

b) y = 2x + 5, y = (1/3)x + 1 => Neither

Graphing a Line Using Intercepts

Steps:

  1. Rearrange the equation of the line into modified standard form.

Standard form: Ax + By + C = 0 => Modified standard form: Ax + By = C

  1. Determine the intercepts.

x-int, y=0: Substitute y = 0 and solve for x.

y-int, x=0: Substitute x = 0 and solve for y.

  1. Plot the intercepts and draw the line.

Finding the Equation of a Line Given Two Points

Recall: The equation of a line can be written in:

  • Slope y-intercept form: y = mx + b
  • Standard form: Ax + By + C = 0

A, B, and C do not provide useful information about the graph. Steps: given (-X1, -Y1) and (X2, -Y2)

  1. Write the format of the equation: y = mx + b. Calculate the slope using the slope formula.

m = (Y2 – Y1) / (X2 – X1)

  1. Rewrite the equation of the line, substituting the value for the slope: y = __x + b.
  2. Choose one coordinate point (x, y) given in the question, substitute x and y into your equation, then solve for b.
  3. Write the equation of the line, replacing only m and b: y = __x + b.

Ex. Find the equation of the line given: a) it passes through (1, 2) set 1 and (5, 10) set 2.

Finding the Equation of a Line Given Slope and One Point

Follow the steps from lesson 5.5 (skip the slope calculation).

Linear Systems

Linear system: A set of two or more linear equations analyzed simultaneously.

Solutions to a linear system – is called the point of intersection (P.O.I) = (x, y)

  • The coordinate point where the lines intersect/cross each other.
  • The point of intersection is the one and only point that both lines share.

Three types of solutions to linear systems:

  1. One solution = point of intersection (x, y) (not working with parallel lines).
  2. No solution: Parallel lines are distinct/different (same slopes, different y-intercepts).
  3. Infinite solutions: Parallel lines are coincident/same (same slope, same y-intercept).

Ex. Solve the linear system and verify the solution.

Solving Linear Systems by Graphing

Gurjot has a budget of $5000 for his birthday party. Which banquet hall offers the better deal, and under what conditions?

Party Palace: $200 (b) for the hall rental plus $40 (m) per guest (y).

Hall for Y’all: $1000 (b) for the hall rental plus $30 (m) per guest (y).

Represent each value: Let n represent the number of people attending (independent variable). Let T represent the total cost (dependent variable).

Party Palace: T = mn + b => T = 40n + 200 (m = 40(x100)/1(x100) = 4000/100; Rise = 4000, Run = 100). 4000 + 200 = 4200

Hall for Y’all: T = mn + b => T = 30n + 1000 (m = 30(x100)/1(x100) = 3000; Rise = 3000, Run = 100). 3000 + 1000 = 4000

Find where the two lines intercept (P.O.I) = (80, 3400). 80 is the number of people; 3400 represents the cost.

  • If Gurjot invites exactly 80 guests, the total cost is the same at both places. It will cost him $3400.
  • If Gurjot has fewer than 80 guests, Party Palace is cheaper (the lower line).
  • If Gurjot has more than 80 guests attending, Hall for Y’all would be the better deal (because the upper dot is lower than the other one).