Problem 25 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((1,2)\) and \((5,10)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points \((1,2)\) and \((5,10)\) is \(y - 2 = 2(x - 1)\) in point-slope form, and \(y = 2x\) in slope-intercept form.

Step 1: Calculate the Slope (m)

The formula to calculate the slope (m), using two points \((x1, y1)\) and \((x2, y2)\), is: \(m = (y2 - y1) / (x2 - x1)\). So, substitute \((x1, y1) = (1,2)\) and \((x2, y2) = (5,10)\) into the formula: \(m = (10-2) / (5-1)\). After doing the subtraction, this becomes: \(m = 8/4 = 2\).

Step 2: Find the Equation in Point-Slope Form

The point-slope form of a linear equation is \(y - y1 = m(x - x1)\). Substitute in \(m = 2\), and choose one of the given points, say, \((x1, y1) = (1, 2)\). The equation becomes \(y - 2 = 2(x - 1)\).

Step 3: Convert the Equation into Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\), where \(b\) is the y-intercept. To convert the equation from step 2 into this form, rearrange the equation: \(y - 2 = 2x - 2\), then simplify this to \(y = 2x\).

Key Concepts

Point-Slope Form

The point-slope form is a convenient way to write the equation of a line when you know the slope and a single point on the line. The general formula is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point on the line, and \(m\) is the slope of the line.

Let us illustrate this with an example from the exercise you are working on. After calculating the slope \(m\) as 2, and given the point (1, 2), you can plug these values into the point-slope formula. Thus, the equation of the line through (1, 2) with a slope of 2 becomes \(y - 2 = 2(x - 1)\).

This version is very useful when only the slope and a single point are known, making it especially helpful in problems where you have a point and the line's inclination but not necessarily the full graph.

Slope-Intercept Form

The slope-intercept form is perhaps the most recognized equation for a straight line, defined as \(y = mx + b\). It clearly shows the slope \(m\) and the y-intercept \(b\), where the line crosses the y-axis.

To convert from point-slope to slope-intercept form, as in your exercise, you take the equation \(y - 2 = 2(x - 1)\) and simplify it. Distribute the slope \(2\) across the \(x - 1\) to get \(2x - 2\). Add 2 to both sides to isolate \(y\), giving you \(y = 2x\) as the final equation of the line in slope-intercept form. Notice in this case, that the line passes through the origin, and hence the y-intercept \(b\) is 0, which is why it's not visible in the final equation.

Slope Calculation

Slope calculation is fundamental when describing the steepness and direction of a line. The slope, denoted as \(m\), can be calculated using two points on a line. The formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), takes the change in y (the rise) over the change in x (the run).

For the points given in your exercise, (1, 2) and (5, 10), plug them into the formula: \(m = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2\). Therefore, the slope of the line is 2, indicating that for every unit increase in x, y increases by 2 units.

Remember that the sign of the slope also indicates the direction of the line. A positive slope, as in this example, means the line is ascending from left to right, while a negative slope means it's descending. Knowing how to calculate the slope is crucial for understanding and writing the equations of lines in different forms.