Problem 26 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((3,5)\) and \((8,15)\)

Step-by-Step Solution

Answer

The line passing through the points \((3,5)\) and \((8,15)\) can be expressed in point-slope form as \(y - 5 = 2(x - 3)\) and in slope-intercept form as \(y = 2x - 1\).

Step 1: Calculate the Slope

The slope of the line passing through \((x_1, y_1) = (3,5)\) and \((x_2, y_2) = (8,15)\) is given by \(m = (y_2 - y_1)/(x_2 - x_1)\). Substituting the given values in, we find that \(m = (15 - 5)/(8 - 3) = 10/5 = 2\).

Step 2: Write in Point-Slope Form

Substitute \(m = 2\), \(x_1 = 3\), and \(y_1 = 5\) into the point-slope form: \(y - y1 = m(x - x1)\) which gives us \(y - 5 = 2(x - 3)\).

Step 3: Write in Slope-Intercept Form

Expand and simplify the equation from step 2, \(y - 5 = 2x - 6\). Add 5 to both sides to isolate \(y\) on one side and obtain the slope-intercept form: \(y = 2x - 1\).

Key Concepts

Slope-Intercept Form

When tackling linear equations, the slope-intercept form is like a trusty compass – it reliably points you in the right direction. This form is written as (y = mx + b), where (m) represents the slope of the line and (b) is the y-intercept. That's where the line crosses the y-axis. Remember, the y-intercept is simply the point where the line taps the vertical axis at (x=0).

The beauty of slope-intercept form lies in its simplicity; it makes graphing straight lines a breeze. Given a slope and an intercept, you can plot the line on a graph with ease. For the math enthusiasts, transforming equations from other forms – like point-slope or standard form – into slope-intercept form is a snap. The goal is to solve for (y) and get everything else neatly arranged on the other side of the equation.

Linear Equations

Linear equations are the straight lines of the math world – simple, direct, and everywhere. They describe a relationship between two variables typically called (x) and (y) where the power of the variables is always 1 - no (x^2)s or square roots here. Their graph is always a line and it extends infinitely in both directions.

The general form of a linear equation is (Ax + By = C), but don't let the letters scare you. You can shuffle them around to get different forms, such as the slope-intercept form already mentioned, or even the point-slope form, which is super handy when you have a point and a slope ready.

Slope Calculation

The slope is the incline, or steepness, of a line, and calculating it is like measuring how quickly a hill rises or falls as you walk along it. To find it, you simply take two points on the line, say ((x_1, y_1)) and ((x_2, y_2)), and use the formula (m = (y_2 - y_1) / (x_2 - x_1)). This gives you the rate at which (y) is changing with respect to (x).

If the result is positive, your line is going up like a sunrise. Negative? It's going down like a sunset. A zero means the road is flat – it's a horizontal line. And if your slope seems like it's going off to infinity, well, that's a vertical line for you; no slope, just infinite possibilities. The slope doesn't just dictate direction but also speed; steeper slopes mean bigger changes in (y) over smaller changes in (x). So, fasten your seatbelt – the higher the number, the steeper the journey.