Linear equations are a cornerstone of algebra, offering insights into relationships between two variables. These equations, when graphed, always result in a straight line. Let’s dive deep into the world of linear equations:
Definition and General Form
A linear equation in two variables typically looks like this:
- y = mx + b
In this equation, y and x are variables. The coefficient m is the slope of the line, indicating the rate at which y changes for every unit change in x. The b represents the y-intercept, which is the value of y when x equals 0.
Key Characteristics
- Straight Line: A linear equation, when plotted, forms a straight line.
- Constant Slope: The slope (m) remains consistent across any two points on the line.
- Intercepts: The line intersects the y-axis at (0, b) and the x-axis where y equals 0.
Solving Linear Equations
Solving these equations involves finding values of y for given x values, or vice versa. This usually requires rearranging the equation to isolate one of the variables.
Examples for Clarity
Example 1:
Consider y = 2x + 3.
- Slope (m): 2
- Y-intercept (b): 3
Choose different x values, calculate corresponding y values, and plot them. For instance, with x = 1, y = 5. The point (1, 5) can be plotted on the graph.
Example 2:
Now, take y = -x + 4.
- Slope (m): -1
- Y-intercept (b): 4
Similarly, select values for x, find y, and plot. If x = 2, y = 2, giving us the point (2, 2).
Real-world Applications
Linear equations find applications in economics for cost modeling, in physics for motion equations, and in daily life activities like budgeting or cooking.
Understanding linear equations is fundamental in mathematics, paving the way for more advanced topics in algebra and calculus.
Linear Equations – Video
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