Linear Equations Solver for Two Variables

What is a Linear Equation with Two Variables?

A linear equation with two variables is like a math puzzle where two unknown numbers work together to make a true statement. It's called "linear" because when we draw it, it makes a straight line!

How to Solve Linear Equations with Two Variables

To solve these equations, we need two of them! We call this a "system of equations." It's like having two clues to find two hidden treasures. We use these clues together to figure out what our unknown numbers (usually called x and y) really are.

Formula

A system of two linear equations with two variables looks like this:

\[ a_1x + b_1y = c_1 \]

\[ a_2x + b_2y = c_2 \]

Where:

• \(x\) and \(y\) are our unknown variables (the treasures we're looking for)
• \(a_1\), \(b_1\), \(a_2\), and \(b_2\) are the coefficients (they tell us how important each variable is)
• \(c_1\) and \(c_2\) are the constants (the numbers that don't change)

Calculation Steps

1. Write out both equations
2. Choose one variable to eliminate (let's say x)
3. Multiply one or both equations so the x terms will cancel out when added or subtracted
4. Add or subtract the equations to eliminate x
5. Solve for y in the resulting equation
6. Substitute this y value into one of the original equations
7. Solve for x
8. Check your answer in both original equations

Example and Visual Representation

Let's solve this system of equations:

\[ 2x + 3y = 13 \]

\[ x + y = 5 \]

We can visualize these equations as lines on a graph:

The blue line represents 2x + 3y = 13, and the red line represents x + y = 5. The green dot where they intersect is our solution: x = 3 and y = 2. You can check that these values satisfy both equations!

This graph shows us that solving linear equations is like finding where two lines meet. It's a bit like a treasure map, where X marks the spot!