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 that passes through the points (1,2) and (5,10) is \(y = 2x\) in slope-intercept form.

Step 1: Calculate the Slope

Firstly, the slope of the line that passes through the points \((1,2)\) and \((5,10)\) needs to be calculated using the formula for slope \[m = (y_2 - y_1) / (x_2 - x_1)\]\nSubstituting the given points into the formula we get: \[m = (10 - 2) / (5 - 1)\]

Step 2: Substitute to Point-Slope Form

After calculating the slope, we substitute one of the given points and the slope into the point-slope form equation, \[y - y_1 = m(x - x_1)\]\nLet's use the point (1,2) to substitute into the equation. \[y - 2 = 2(x - 1)\]

Step 3: Derive the Slope-Intercept Form

From the point-slope form obtained in step 2, we derive the slope-intercept form (y = mx + b). This requires making y the subject of the formula and simplifying.\n\[y = 2x - 2 + 2\]\nSo, in slope-intercept form, the equation of the line is \[y = 2x\]

Key Concepts

Point-Slope Form

When tackling linear equations, understanding the point-slope form is crucial. This form of a line's equation is expressed as \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) represents a point through which the line passes, and \(m\) stands for the slope of the line.

Using point-slope form is particularly helpful when you know:

• One point on the line
• The slope of the line

For example, if you have points like \((1,2)\) and \(m = 2\), you can quickly write the equation of the line: \( y - 2 = 2(x - 1) \). In this formula, \(x_1 = 1\) and \(y_1 = 2\), readily substitutable into the expression to capture the line's dynamics. This form readily shows how any given point balances around the slope and its position on the line.

Point-slope is convenient for building expressions step-by-step, especially useful in further deriving other forms like the slope-intercept form.

Slope-Intercept Form

The slope-intercept form of a linear equation is often one of the clearest ways to express a line in algebra. It is given by the formula \( y = mx + b \). In this format:

• \(m\) represents the slope of the line
• \(b\) is the y-intercept, where the line crosses the y-axis

The simplicity of this form lies in how intuitive it is to locate and interpret a line's graph. When you have this form, you can easily recognize both the steepness of the line (the slope \(m\)) and where it meets the y-axis (at \(b\)).

If you derived the following from your point-slope conversion: \( y = 2x \), it implies that the line goes through the origin \((0,0)\), because \(b = 0\). This means no vertical shift, just slope. From understanding the relationship between \(m\) and \(b\), readers can instantly plot clear, understandable graphs of linear equations.

Slope Calculation

Calculating the slope is a foundational skill when dealing with linear equations. The slope measures how steep a line is and is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here's how to effectively compute it:

• Identify two points on the line, say \((x_1, y_1)\) and \((x_2, y_2)\)
• Substitute these points into the formula

For example, given points \((1,2)\) and \((5,10)\), you find the slope by calculating \( m = \frac{10 - 2}{5 - 1} = 2 \). A slope of 2 means that for every unit increase in the x-direction, the y value increases by 2 units, indicating a rising line.

Understanding the slope helps predict the course of a line across a graph and plays an essential role in defining equations uniquely. By mastering slope calculations, students can manipulate and transform various forms of a line effectively.