Linear Equations-In-Two-Variables

Linear Equations in Two Variables

In this chapter, you will learn about linear equations in two variables, how to represent them, solve them, and use them in real-life situations. By the end of this chapter, you will be able to write, solve, and graph linear equations in two variables.

Key Concepts

• Linear Equation: An equation of the form ax + by + c = 0, where a, b, and c are real numbers and x, y are variables.
• Solution: A pair of values (x, y) that satisfy the equation.
• Graph: The set of all solutions plotted on a coordinate plane forms a straight line.

Standard Form

The standard form of a linear equation in two variables is ax + by + c = 0 or ax + by = c.

• Examples:
  
  • 2x + 3y = 6
  • x - y + 4 = 0
  • 5x + 2y = 10

Solution of a Linear Equation

Any pair of values (x, y) that makes the equation true is called a solution.

• Example: For the equation x + y = 5, (2, 3) and (1, 4) are solutions because 2 + 3 = 5 and 1 + 4 = 5.

Graphing Linear Equations

The graph of a linear equation in two variables is a straight line. To draw the graph:

• Find at least two solutions (pairs of x and y).
• Plot these points on the coordinate plane.
• Draw a straight line through the points.

Applications

• Solving word problems involving two unknowns.
• Representing relationships between quantities.
• Finding intersection points of two lines (solutions to two equations).

Practice Questions

1. Write the standard form of the equation: y = 2x + 3.
2. Find two solutions for the equation: x + y = 7.
3. Plot the graph of the equation: x - y = 2.
4. If 3x + 2y = 12, find the value of y when x = 2.
5. Explain why the graph of a linear equation is a straight line.

Challenge Yourself

• Find the point where the lines x + y = 6 and x - y = 2 intersect.
• Write a word problem that can be solved using a linear equation in two variables.

Did You Know?

• Every straight line on a graph can be represented by a linear equation in two variables.
• Linear equations are used in business, science, and engineering to solve real-world problems.

Glossary

• Variable: A symbol (like x or y) that stands for a number.
• Equation: A statement that two expressions are equal.
• Coordinate Plane: A grid for plotting points using x and y values.

Answers to Practice Questions

1. y - 2x = 3 or 2x - y + 3 = 0
2. (0, 7), (3, 4), (5, 2) (any two pairs that add up to 7)
3. Plot points like (2, 0) and (4, 2), then draw a line through them.
4. 3x + 2y = 12 → 3(2) + 2y = 12 → 6 + 2y = 12 → 2y = 6 → y = 3
5. Because every solution forms a point, and all points together make a straight line.

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Practice solving and graphing linear equations to master this important topic!