Problem 31 - Prealgebra

Question

Graph the points \((-3,2)\) and \((1,3),\) and draw a line through them. Use that graph to answer Problems 31-34. Does the ordered pair ( \(5,4\) ) lie on this line? (GRAPH CANT COPY)

Step-by-Step Solution

Answer

Yes, the point \((5, 4)\) is on the line.

Step 1: Understand the Slope Formula

To draw a line through the points \((-3, 2)\) and \(1, 3)\), we first need to calculate the slope of the line connecting them. The slope \(m\) of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as \{ m = \frac{y_2 - y_1}{x_2 - x_1} \}.

Step 2: Calculate the Slope

Using the slope formula from Step 1, substitute the coordinates: \( (x_1, y_1) = (-3, 2) \) and \( (x_2, y_2) = (1, 3) \).\[ m = \frac{3 - 2}{1 + 3} = \frac{1}{4} \]

Step 3: Determine the Line Equation

Using the slope-intercept form \( y = mx + b \), where \(m = \frac{1}{4}\), substitute one of the points to find \(b\). Using \((-3, 2)\): \[ 2 = \frac{1}{4}(-3) + b \]Solve for \(b\):\[ 2 = -\frac{3}{4} + b \\frac{11}{4} = b \] Thus, the equation of the line is \( y = \frac{1}{4}x + \frac{11}{4} \).

Step 4: Check if \((5, 4)\) is on the Line

Substitute \(x = 5\) into the line equation and see if it results in \(y = 4\):\[ y = \frac{1}{4}(5) + \frac{11}{4} = \frac{5}{4} + \frac{11}{4} = \frac{16}{4} = 4 \]Since the calculation gives \(y = 4\), the point \((5, 4)\) lies on the line.

Key Concepts

Slope Formula

The slope formula is a key component in coordinate geometry, helping us understand how steep a line is on a graph. The slope, usually represented by the letter \(m\), indicates the change in the vertical direction (rise) for every unit of change in the horizontal direction (run). You can calculate the slope of a line passing through any two points \((x_1, y_1)\) and \((x_2, y_2)\) using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula essentially tells you how much \(y\) changes for a unit change in \(x\). A positive slope means the line is rising as it moves from left to right, while a negative slope means the line is falling. A slope of zero indicates a perfectly horizontal line, and an undefined slope (when the denominator \(x_2 - x_1 = 0\)) refers to a vertical line.

Graphing Points

Graphing points is the first step in understanding their geometric relationship on a plane. Each point is defined by an \((x, y)\) coordinate that tells you exactly where it is located on the graph. For example, the point \((-3, 2)\) means you move 3 units left of the origin on the x-axis and 2 units up on the y-axis. Similarly, the point \((1, 3)\) means you move 1 unit to the right and 3 units up.Once you have plotted points like \((-3, 2)\) and \((1, 3)\), connecting them with a straight line reveals the direction and steepness of the slope. Always ensure precision by using a ruler or straight edge to help draw the line cleanly through the plotted points. Graphing is a visual method to understand relationships, and once points are plotted, they can easily indicate what other points lie on the same line.

Equation of a Line

The equation of a line connects algebra and geometry by describing all the points lying on a line. The most common form is the slope-intercept form, written as \(y = mx + b\), where:

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

This equation makes it simple to graph a line because you start at the y-intercept \(b\), and use the slope \(m\) to identify another point on the line. For the line passing through \((-3, 2)\) and \((1, 3)\), we calculated \(m = \frac{1}{4}\) and found the y-intercept \(b = \frac{11}{4}\). Therefore, the line equation becomes \(y = \frac{1}{4}x + \frac{11}{4}\).To confirm whether a point like \((5, 4)\) is on this line, substitute \(x = 5\) into the equation and see if it results in \(y = 4\). This substitution step verifies if the coordinates satisfy the equation, confirming the point's position on the line.