Write a linear equation from a slope and a point
If you know a line's slope and just one point it passes through, you have enough information to write its equation. This is a powerful tool for modeling linear relationships in the real world.
Concept
The most common form of a linear equation is the slope-intercept form: **y = mx + b**. * **m** is the slope. * **b** is the y-intercept (the point where the line crosses the y-axis). To write an equation from a slope and a point, you need to find the value of 'b'. **The Plug-in Method** 1. **Start** with the slope-intercept formula: y = mx + b. 2. **Plug in** the slope (m) and the coordinates of the given point (x, y). 3. **Solve** the resulting equation for 'b'. 4. **Write** the final equation with the values for m and b. **Example: Write the equation for a line that has a slope of 2 and passes through the point (3, 7).** 1. **Start with the formula:** y = mx + b 2. **Plug in the values:** We know m = 2, x = 3, and y = 7. * 7 = 2(3) + b 3. **Solve for b:** * 7 = 6 + b * Subtract 6 from both sides: 7 - 6 = b * b = 1 4. **Write the final equation:** Now we know m = 2 and b = 1. * **y = 2x + 1** **Point-Slope Form (An Alternative Method)** Another useful form of a linear equation is the point-slope form: **y - y₁ = m(x - x₁) ** * You just plug in the slope 'm' and the point (x₁, y₁). **Example (Same as above): slope = 2, point = (3, 7)** 1. **Start with the formula:** y - y₁ = m(x - x₁) 2. **Plug in the values:** m = 2, x₁ = 3, y₁ = 7. * y - 7 = 2(x - 3) 3. **(Optional) Convert to slope-intercept form:** * Distribute the 2: y - 7 = 2x - 6 * Add 7 to both sides: y = 2x + 1 Both methods give the same result! **Key Idea:** Every point on a line is a solution to its equation. That's why you can plug in the x and y from any point on the line to help you find the missing piece of the equation (the y-intercept 'b').
Try it
Practice: Write a linear equation from a slope and a point.