The point-slope form is a mathematical method used to define the equation of a straight line. It is particularly useful when you know at least one specific point on the line and the steepness (slope) of that line. This form allows you to quickly visualize the line's direction and position without needing the y-intercept immediately.
The Point-Slope Formula
y – y₁ = m(x – x₁)
m represents the slope of the line.
(x₁, y₁) represents the coordinates of the known point.
x and y are the variables representing any other point on the line.
Converting to Other Forms
While the point-slope form is great for writing equations quickly, you often need to convert it into other formats:
Slope-Intercept Form (y = mx + b): To find this, distribute the slope m through the parentheses and solve for y. The b represents the point where the line crosses the y-axis.
Standard Form (Ax + By = C): This is used for linear programming and solving systems of equations. In this form, A, B, and C are typically integers.
Practical Example
Suppose you have a line with a slope of 2 that passes through the point (3, 4).
Point-Slope: y – 4 = 2(x – 3)
Slope-Intercept: y = 2x – 2
Standard Form: -2x + y = -2 (or 2x – y = 2)
function calculatePointSlope() {
var m = parseFloat(document.getElementById('slopeInput').value);
var x1 = parseFloat(document.getElementById('x1Input').value);
var y1 = parseFloat(document.getElementById('y1Input').value);
var resultDiv = document.getElementById('psf-result');
if (isNaN(m) || isNaN(x1) || isNaN(y1)) {
alert("Please enter valid numbers for slope and coordinates.");
return;
}
// 1. Point Slope Form String
var xPart = x1 < 0 ? "+ " + Math.abs(x1) : "- " + x1;
var yPart = y1 = 0 ? "+ " : "- ";
var siEquation = "y = " + m + "x " + bSign + Math.abs(b).toFixed(2).replace(/\.00$/, "");
if (b === 0) siEquation = "y = " + m + "x";
document.getElementById('slopeInterceptResult').innerText = siEquation;
document.getElementById('yInterceptResult').innerText = b.toFixed(2).replace(/\.00$/, "");
// 3. Standard Form (Ax + By = C)
// -mx + y = b
var A = -m;
var B = 1;
var C = b;
// Formatting Standard Form to look clean (Ax + By = C)
var aStr = A === -1 ? "-x" : (A === 1 ? "x" : A + "x");
var bStr = B >= 0 ? "+ " + B + "y" : "- " + Math.abs(B) + "y";
if (B === 1) bStr = "+ y";
if (B === -1) bStr = "- y";
var stEquation = aStr + " " + bStr + " = " + C.toFixed(2).replace(/\.00$/, "");
document.getElementById('standardFormResult').innerText = stEquation;
resultDiv.style.display = 'block';
}