Problem 23 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line passing through the two points. Show that the line is perpendicular to the given line. Check your answer by graphing both lines. $$ (4,-7),(7,5) ; y=-\frac{1}{4} x $$

Step-by-Step Solution

Answer

The line passing through the points (4,-7) and (7,5) is \(y=4x-23\) which is perpendicular to the line \(y=-\frac{1}{4} x\). This can be verified by their graph.

Step 1: Calculate the slope (m) of the line passing through the points (4,-7) and (7,5)

The slope between two points can be calculated using the formula \(m= \frac{y_2-y_1}{x_2-x_1}\).\nUsing this, the slope, \(m\), between the two points is given by \(m = \frac{5-(-7)}{7-4} = 4\)

Step 2: Write the equation of the line in slope-intercept form

The equation of a line in slope-intercept form is \(y = mx + c\), where m represents the slope and c is the y-intercept. Now, substitute m=4 from step one into the equation and one of the points into the equation to find c. \nUsing the point (4,-7), the equation becomes \n-7 = 4 * 4 + c \n On simplifying this, we find that c=-23

Step 3: Check if the line is perpendicular to \(y=-\frac{1}{4} x\)

Two lines are perpendicular if and only if the product of their slopes is -1. From the equation given, the slope of the second line is -\frac{1}{4}. The slope of our line from step 1 is 4. The product of these slopes is \(4*(-\frac{1}{4})=-1\), which confirms that the line is indeed perpendicular to \(y=-\frac{1}{4} x\)

Step 4: Graph the lines

Plot the line from Step 2, \(y=4x-23\), and \(y=-\frac{1}{4} x\) on the same graph sheet. The intercepts and slope of these functions will give us the respective lines. If the lines drawn are perpendicular then it implies that the answer is correct.

Key Concepts

Perpendicular Lines

When discussing lines in geometry, two lines are said to be perpendicular if they intersect at a 90-degree angle. This is a fundamental concept in both geometry and algebra. The key property to identify perpendicular lines in a coordinate plane is to look at their slopes. Specifically:

• Two lines are perpendicular if the product of their slopes equals -1.
• This means that if one line has a slope of \( m \), then the line perpendicular to it will have a slope of \( -\frac{1}{m} \). This is because the slopes are negative reciprocals of each other.

In our case, we are given a line with the equation \( y = -\frac{1}{4}x \), which means the slope is \(-\frac{1}{4}\). To check if the line through the points \((4, -7)\) and \((7, 5)\) is perpendicular to this line, we calculated its slope to be \(4\). Multiplying these slopes, \(4 \times -\frac{1}{4}\), results in \(-1\), confirming that the lines are indeed perpendicular. Remember, understanding perpendicularity can help in various applications, from designing right angles to verifying work in both mathematics and real-world projects.

Graphing Lines

Graphing lines involves plotting the linear equation on a coordinate plane. This requires understanding the slope-intercept form of the equation, \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

• The y-intercept \( c \) indicates where the line crosses the y-axis, meaning that the point \((0, c)\) is on the line.
• The slope \( m \) tells you how to move from one point to another along the line. A positive slope means moving up and to the right, whereas a negative slope suggests moving down and to the right.

Using these concepts, we can easily draw any given line. For example, the equation \( y = 4x - 23 \) tells us:- The slope \(4\) means for every increase of \(1\) in \(x\), \(y\) increases by \(4\).- The y-intercept \(-23\) is the starting point on the y-axis. Plot the intercept first, and then use the slope to find another point. Repeat this process to extend and draw the line on a graph. Being able to graph lines is an essential skill for visualizing data, analyzing trends, and solving geometric problems.

Equation of a Line

The equation of a line is a mathematical description of a linear relationship between two variables, \(x\) and \(y\). It is usually expressed in the slope-intercept form: \( y = mx + c \).

• Slope \( m \):The slope indicates the rate of change. It tells us how much the \(y\)-value changes for a unit change in \(x\). A steeper line has a larger absolute slope value.
• Y-Intercept \( c \):This is where the line crosses the y-axis. The y-intercept is the value of \(y\) when \(x = 0\).

For example, consider the line equation derived from the points \((4, -7)\) and \((7, 5)\). We first calculate the slope \( m = 4 \) using the formula \( \frac{y_2 - y_1}{x_2 - x_1} \). Then, using one of the points, such as \((4, -7)\), we substitute into the slope-intercept form to solve for the y-intercept \( c \). This gives us the full equation \( y = 4x - 23 \). Knowing how to derive and use the equation of a line is useful for predicting outcomes, analyzing relationships, and solving algebraic equations.