Problem 33 - Introductory Algebra for College Students

Question

Write an equation in slope-intercept form of the line satisfying the given conditions. The line passes through \((4,-7)\) and is perpendicular to the line whose equation is \(x-2 y=3.\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -2x + 1\).

Step 1: Finding the Slope of the Given Line

First, we need to express the equation of the given line (\(x-2y=3\)) in slope-intercept form to find its slope. We can do this by isolating y in terms of x. \n We add 2y to both sides and subtract 3 from both sides to get \(x-3=2y\), which simplifies to \(y=\frac{x}{2}-\frac{3}{2}\). Given that the slope-intercept form is \(y=mx+b\), we can see that the slope (m) of the given line is \(\frac{1}{2}\).

Step 2: Finding the Slope of the Perpendicular Line

The line being searched for is perpendicular to the given line. By rule, the slope of a line perpendicular to another is the negative reciprocal of the other line's slope. So, the slope (m) of the line being searched for will be the negative reciprocal of \(\frac{1}{2}\), which is -2 (\(-\frac{1}{\frac{1}{2}} = -2\)).

Step 3: Determine the Y-Intercept of the Line

Now, we use the point-slope formula which is \(y-y_1 = m(x-x_1)\), where \((x_1, y_1)\) is a given point on the line (which is (4,-7) in the case of our problem) and m is our newly found slope. We plug in these values and simplify to get \(y+7 = -2(x-4)\). Simplifying further, we get \(y = -2x + 8 - 7\), which simplifies to \(y = -2x + 1\). This equation represents the line perpendicular to the given line and passing through the point \((4,-7)\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations. It's a simple and useful formula: \(y = mx + b\). In this formula:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope of the line, showing how steep the line is.
• \(b\) is the y-intercept, where the line crosses the y-axis.

To convert a linear equation into slope-intercept form, solve for \(y\) so it stands alone on one side of the equation. This form makes it easy to quickly identify the slope and y-intercept. Understanding these elements helps in graphing the line or finding other characteristics such as parallel or perpendicular lines.

Perpendicular Lines

When discussing perpendicular lines in terms of slope, there is an important rule to remember: two lines are perpendicular if the product of their slopes is -1. Simply put, if one line has a slope of \(m\), the line perpendicular to it will have a slope of \(-1/m\). This relationship ensures that the lines intersect at a right angle.
For example, if you know that a line has a slope of \(1/2\), a line perpendicular to it will have a slope of \(-2\) because \( \left( \frac{1}{2} \right) \times (-2) = -1 \).
This is very useful when you need to find an equation for a line perpendicular to another, as in many math and geometry problems.By using the reciprocal and changing the sign, you maintain the perpendicular relationship.

Point-Slope Formula

The point-slope formula is another tool for writing equations of lines. It is particularly useful when you know one point on the line and its slope. The formula is: \(y-y_1 = m(x-x_1)\)

• \((x_1, y_1)\) is a point on the line.
• \(m\) is the slope of the line.

To use the point-slope formula, plug in the values for a known point you have and the slope. This will allow you to easily find other points that lie on the line or convert the equation into the slope-intercept form for graphing.
This method is especially handy when solving problems that require you to form an equation using a specific point, as is common in many exercises on geometry and algebra.

Linear Equations

Linear equations are foundational in algebra. They represent straight lines in the coordinate plane. The general form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
These equations can be converted into different forms such as the slope-intercept form, which enables different kinds of analysis or graphing.Linear equations are simply about the connection between two variables, with each equation forming a straight line when graphed.

• They can have one or more solutions.
• They can represent real-world relationships, like distance over time or cost per item.

Grasping linear equations is crucial for more complex algebraic and geometric concepts, as these are the building blocks for understanding the behavior of lines on a graph.