Linear Equations: Two Variables

Linear Equations in Two Variables

• Equations are written in the form (ax + by + c = 0).
• (a), (b), and (c) represent real numbers.
• Both (a) and (b) cannot be zero.

Standard Form

• (ax + by + c = 0) is the standard form of a linear equation in two variables.

Solutions of Linear Equations

• Solutions are pairs of values for (x) and (y) that satisfy the equation.
• Each solution corresponds to a point on the line representing the equation.

Graphing Linear Equations

• Graphing involves plotting the line that represents all solutions to the equation.
• At least two points satisfying the equation are needed to graph it.
• Plot the points on a coordinate plane and draw a straight line through them.
• The line visually represents all possible solutions.

Slope-Intercept Form

• Expressed as (y = mx + b).
• (m) is the slope.
• (b) is the y-intercept.
• The slope (m) indicates the rate of change of (y) with respect to (x).
• The y-intercept (b) is where the line crosses the y-axis.

Finding the Slope and Intercept

• Given two points ((x_1, y_1)) and ((x_2, y_2)), the slope (m) is calculated as (m = \frac{y_2 - y_1}{x_2 - x_1}).
• The y-intercept is found by substituting the slope and one point into (y = mx + b) and solving for (b).

Parallel and Perpendicular Lines

• Parallel lines share the same slope.
• Perpendicular lines have slopes that are negative reciprocals, so (m_1 \cdot m_2 = -1).

Systems of Linear Equations

• Consist of two or more linear equations with the same variables.
• The solution is the set of values for the variables that satisfy all equations.

Methods to Solve Systems of Equations

• Substitution: Solve one equation for one variable and substitute into the other.
• Elimination: Add or subtract equations to eliminate one variable.
• Graphical: Graph both equations and find the intersection point.

Substitution Method

• Solve one equation for one variable.
• Substitute the expression into the other equation.
• Solve for the remaining variable.
• Substitute the value back into one of the original equations to find the other variable’s value.

Elimination Method

• Multiply equations by constants so that one variable's coefficients are equal or opposite.
• Add or subtract the equations to eliminate one variable.
• Solve for the remaining variable.
• Substitute the value back into an original equation to find the eliminated variable’s value.

Graphical Method

• Graph each equation on the same coordinate plane.
• Identify the point of intersection.
• The intersection point’s coordinates are the solution to the system.

Types of Solutions

• Unique Solution: Lines intersect at one point.
• No Solution: Lines are parallel and do not intersect.
• Infinitely Many Solutions: Lines are identical and overlap.

Applications in Real-World Scenarios

• Used to model relationships between two variables.
• Represent quantities that change at a constant rate.
• Applications include cost analysis, distance-rate-time relationships, and mixture problems.

Cost Analysis

• Models the relationship between cost and quantity.
• Can represent total manufacturing cost with a fixed setup cost and a variable cost per item.

Distance-Rate-Time Relationships

• Modeled using the formula (d = rt).
• (d) represents distance, (r) represents rate (or speed), and (t) represents time.

Mixture Problems

• Combine substances with different concentrations.
• Linear equations determine the amount of each substance needed.

Problem Solving Strategy

• Read the problem and identify knowns and unknowns.
• Assign variables to the unknowns.
• Write a system of linear equations representing the problem.
• Solve the system using substitution, elimination, or graphing.
• Interpret the solution in the context of the problem.

Example: Cost Analysis

• A company makes t-shirts with a fixed cost of $500 and a variable cost of $8 per shirt, selling them for $12 each
• To determine the break-even point, (x) is the number of t-shirts
• Total cost is (C = 500 + 8x)
• Total revenue is (R = 12x)
• Set (C = R):
  • (500 + 8x = 12x)
  • (500 = 4x)
  • (x = 125)
• The break-even point is 125 t-shirts.

Example: Distance-Rate-Time

• Two cars start at the same point, traveling in opposite directions at 60 mph and 50 mph
• To find how long until they are 330 miles apart, (t) is the time in hours
• The first car's distance is (d_1 = 60t)
• The second car's distance is (d_2 = 50t)
• Total distance is (d_1 + d_2 = 330)
  • (60t + 50t = 330)
  • (110t = 330)
  • (t = 3)
• It will take 3 hours for the cars to be 330 miles apart.

Example: Mixture Problem

• A chemist mixes a 20% acid solution with a 50% acid solution to get 100 ml of a 30% acid solution.
• Let (x) be the amount of 20% solution and (y) the amount of 50% solution.
• Total volume: (x + y = 100)
• Amount of acid: (0.20x + 0.50y = 0.30(100))
• Solving the system:
  • (x + y = 100)
  • (0.20x + 0.50y = 30)
• From the first equation, (y = 100 - x)
• Substitute:
  • (0.20x + 0.50(100 - x) = 30)
  • (0.20x + 50 - 0.50x = 30)
  • (-0.30x = -20)
  • (x = \frac{20}{0.30} = \frac{200}{3} \approx 66.67) ml
• (y = 100 - x = 100 - \frac{200}{3} = \frac{100}{3} \approx 33.33) ml
• Approximately 66.67 ml of the 20% solution and 33.33 ml of the 50% solution are needed.

Concepts and Important Points

• Systems can have one, none, or infinitely many solutions.
• Verify solutions by substituting values back into the original equations.
• Define variables and set up equations carefully in word problems.
• Linear equations are fundamental in algebra with applications in science, engineering, economics, and computer science.