Equation of a straight line: y=mx+c
Introduction to the Equation of a Straight Line: y=mx+c
- The equation of a straight line in the Cartesian coordinate system is given by y = mx + c.
- Each letter in this equation has a particular meaning: y is the dependent variable, x is the independent variable, m is the gradient (slope) of the line, and c is the y-intercept.
- The letter m represents the gradient of the line, which is the measure of how steep the line is.
- The y-intercept, represented by c, is the point on the graph where the line crosses the y-axis.
Finding the Gradient and y-intercept
- The gradient (m) of the line can be found by taking any two points on the line, (x1, y1) and (x2, y2), and calculating the change in y divided by the change in x: m = (y2 - y1) / (x2 - x1).
- The y-intercept (c) can be found by rearranging the equation to c = y - mx, and subbing in any point on the line for x and y.
Drawing and Interpreting the Line
- When given the equation y = mx + c, you can draw the line on a graph by first marking the y-intercept (c) on the y-axis and then using the gradient (m) to find another point on the line.
- For example, if the gradient is 2, for every step you move right on the x-axis, you move up two steps on the y-axis. If the gradient is -2, for every step you move right, you move down two steps.
- The gradient also indicates the direction of the line: a positive gradient means the line slopes upwards from left to right, while a negative gradient means it slopes downwards.
Examples
- For instance, consider the equation y = 3x + 2. Here, the gradient (m) is 3 and the y-intercept (c) is 2. So, the line cuts the y-axis at 2 and for every 1 unit increase in x, y increases by 3 units.
- Another example could be y = -2x + 1. Here, the gradient is -2 and the y-intercept is 1. The line intersects the y-axis at 1 and for every 1 unit increase in x, y decreases by 2 units.
Conclusion
- Understanding the equation of a straight line y=mx+c is crucial in algebra, as it provides the basis for exploring relationships between variables.
- The gradient and the y-intercept play a crucial role in defining the line’s characteristics. Hence, it’s essential to understand how to find them and what they represent.
- Practice is key for comprehension and confidently applying knowledge of this equation in a range of mathematical problems.