Problem 51 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the two points. $$ (1,3),(2,5) $$

Step-by-Step Solution

Answer

The point-slope form of the equation of the line that passes through the points (1,3) and (2,5) is \(y - 3 = 2(x - 1)\).

Step 1: Calculate the slope of the line

To calculate the slope of the line, use the formula \(m = (y2 - y1) / (x2 - x1)\), substituting the given points (1, 3) for \((x1, y1)\) and (2, 5) for \((x2, y2)\). Therefore, \(m = (5 - 3) / (2 - 1) = 2 / 1 = 2\).

Step 2: Write the point-slope form of the equation

The point-slope form of the equation of a line is \(y - y1 = m(x - x1)\). So, using the calculated slope (m = 2) and one of the given points (for instance, (1, 3)), the equation is \(y - 3 = 2(x - 1)\).

Key Concepts

Slope Calculation

Calculating the slope of a line is a fundamental step in understanding linear equations. The slope is a measure of the steepness or incline of a line. It tells you how much the line rises or falls as you move from one point to another, horizontally. To calculate the slope, use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where

• \(y_1\) and \(y_2\) are the y-coordinates of two points on the line
• \(x_1\) and \(x_2\) are the x-coordinates of these points

For example, with points (1, 3) and (2, 5), calculate the slope as follows:

• \(x_1 = 1\), \(y_1 = 3\)
• \(x_2 = 2\), \(y_2 = 5\)

Substitute these into the formula to get the slope: \( m = \frac{5 - 3}{2 - 1} = 2 \). This means for every unit increase in x, the y-value increases by 2.

Equation of a Line

Once you have the slope of a line, you can create its equation using different forms. A common form is the point-slope form, which is especially useful when you know one point on the line and the slope. The point-slope form equation is: \( y - y_1 = m(x - x_1) \) Here,

• \(m\) is the slope of the line
• \(x_1\) and \(y_1\) are the coordinates of a known point on the line

For instance, if you know the slope \(m = 2\) and a point (1, 3), the equation of the line becomes: \( y - 3 = 2(x - 1) \). This equation can then be rearranged into other forms, like the slope-intercept form, if needed.

Coordinates

Coordinates provide a system for identifying the location of points on a plane, using an ordered pair, usually in the form \((x, y)\). Coordinates tell us the exact position of a point by defining its horizontal and vertical displacements from a standard reference point known as the origin (0, 0).

• The first number, \(x\), is the horizontal value (x-coordinate)
• The second number, \(y\), is the vertical value (y-coordinate)

In the exercise, you have the points (1, 3) and (2, 5). Here, (1, 3) means the point is 1 unit along the x-axis and 3 units up the y-axis. Similarly, (2, 5) is 2 units along the x-axis and 5 units up the y-axis. Knowing how to plot and use these coordinates is key in graphing lines and solving related problems.

Linear Equations

Linear equations represent straight lines on a coordinate plane and can be formed in several ways including slope-intercept, point-slope, and standard forms. The common features of linear equations are their constant rate of change, which means they have a constant slope, and their representation using the variables \(x\) and \(y\).
Linear equations can be used to model real-life situations where change is constant. They take the general form: \( y = mx + b \),

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

In our exercise, starting with the point-slope form equation, \( y - 3 = 2(x - 1) \), you can easily transform to slope-intercept form (\( y = 2x + 1 \)) by solving for \( y \). This highlights how flexible linear equations can be in representing different aspects of a line.