In coordinate geometry, every straight line can be represented by an equation. When we're given a point like (1, 2), we can draw many different lines that pass through it. To write a unique equation for a line, we need additional information such as the slope of the line.
The point-slope form of a line equation is y minus y one equals m times x minus x one. Here, x one and y one are the coordinates of a known point on the line, and m is the slope. For example, if we have the point (1, 2) and a slope of 1, we substitute these values into the formula to get y minus 2 equals 1 times x minus 1.
To convert from point-slope form to slope-intercept form, we solve for y. Starting with y minus 2 equals 1 times x minus 1, we first distribute the 1 to get y minus 2 equals x minus 1. Then we add 2 to both sides to isolate y, resulting in y equals x plus 1. In this form, we can easily identify the y-intercept, which is the point where the line crosses the y-axis.
When we have two points instead of a point and slope, we first calculate the slope using the formula m equals y two minus y one over x two minus x one. For example, with points (1, 2) and (2, 3), we substitute into the formula to get 3 minus 2 over 2 minus 1, which equals 1. Once we have the slope, we can use the point-slope form with either point to write the equation.
The standard form of a line equation is Ax plus By equals C, where A, B, and C are integers and A is typically positive. To convert from slope-intercept form to standard form, we move all variables to one side. Starting with y equals x plus 1, we subtract x from both sides to get negative x plus y equals 1. To make the coefficient of x positive, we multiply the entire equation by negative 1, resulting in x minus y equals negative 1. This form is particularly useful for finding intercepts quickly.
There are two special cases of lines we need to consider. Horizontal lines have equations of the form y equals b, where b is the y-coordinate of all points on the line. These lines have a slope of zero. Vertical lines have equations of the form x equals a, where a is the x-coordinate of all points on the line. These lines have an undefined slope and cannot be written in slope-intercept form because they do not pass the vertical line test for functions.
Let's work through a complete example. Given the point (1, 3) and slope m equals 2, we use the point-slope form: y minus 3 equals 2 times x minus 1. Distributing the 2 gives us y minus 3 equals 2x minus 2. Adding 3 to both sides yields the slope-intercept form: y equals 2x plus 1. We can verify this by substituting x equals 1 into the equation, which gives y equals 2 times 1 plus 1 equals 3, confirming our point (1, 3) is on the line.
To efficiently write line equations, follow this problem-solving strategy. First, identify what information you're given. If you have a point and slope, use the point-slope form directly. If you have two points, calculate the slope first using the slope formula, then use point-slope form. Check for special cases like horizontal or vertical lines. Next, choose the appropriate form based on your needs - point-slope for quick writing, slope-intercept for graphing, or standard form for certain applications. Then substitute your known values into the formula. Finally, simplify the equation to your desired form, making sure coefficients are integers in standard form with a positive leading coefficient.