Problem 65 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$y=4 x$$

Step-by-Step Solution

Answer

The coordinates are (1, 4) and (2, 8) with a slope of 4.

Step 1: Understanding the Line Equation

The given equation is \(y = 4x\). This is the equation of a straight line in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = 4\) and \(b = 0\).

Step 2: Choose a Value for x to Find a Point

Let's choose \(x = 1\) to find a point on the line. Substitute \(x = 1\) into the equation \(y = 4x\) to find \(y\). Thus, \(y = 4(1) = 4\). So, one point on the line is \((1, 4)\).

Step 3: Choose Another Value for x to Find a Second Point

Let's choose \(x = 2\) to find another point on the line. Substitute \(x = 2\) into the equation \(y = 4x\) to find \(y\). Thus, \(y = 4(2) = 8\). So, another point on the line is \((2, 8)\).

Step 4: Use the Two Points to Find the Slope

Use the coordinates \((x_1, y_1) = (1, 4)\) and \((x_2, y_2) = (2, 8)\) to find the slope using the formula: \ \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] \ Substituting the values, we get: \ \[ m = \frac{8 - 4}{2 - 1} = \frac{4}{1} = 4 \] \ Thus, the slope of the line is 4.

Key Concepts

Line Equation

A line equation is a mathematical way to describe a straight line on a graph. The general formula for a line equation given in slope-intercept form is \( y = mx + b \), where:

• \( y \) represents the output or dependent variable.
• \( x \) is the input or independent variable.
• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In this particular exercise, the equation provided is \( y = 4x \), meaning \( m = 4 \) and \( b = 0 \). This tells us that for every unit increase in \( x \), \( y \) increases by 4 units. Since the y-intercept \( b = 0 \), the line passes through the origin (0, 0) on the graph.

Coordinates

Coordinates are used to specify the exact location of a point on a graph. Typically presented as \((x, y)\), coordinates are a pair of numbers:

• The first number \( x \) is the horizontal position on the graph.
• The second number \( y \) is the vertical position on the graph.

For the equation \( y = 4x \), you can find coordinates by selecting a value for \( x \) and calculating the corresponding \( y \). As shown in the exercise, choosing \( x = 1 \) gives the point \((1, 4)\) and choosing \( x = 2 \) gives \((2, 8)\). These are two specific locations of the line on the graph that confirm it is straight and follows the rule set by the line equation.

Slope-Intercept Form

The slope-intercept form of a line equation, \( y = mx + b \), is highly useful for quickly identifying key characteristics of the line without graphing it. Here:

• The slope \( m \) tells us the tilt or steepness of the line. For example, in \( y = 4x \), the slope \( m = 4 \) means the line rises 4 units for every 1 unit it moves to the right.
• The y-intercept \( b \) provides the starting point on the y-axis, where \( x = 0 \). In our example, \( b = 0 \), so the line goes through the origin.

Knowing the form makes it effortless to construct graphs or understand how changes to \( m \) or \( b \) will, in turn, alter the line. This makes slope-intercept form dynamite for both simple and more complex algebraic tasks.