Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Write an equation in slope-intercept form of the line satisfying the given conditions. The line passes through \((-5,6)\) and is perpendicular to the line that has an \(x\) -intercept of 3 and a \(y\) -intercept of \(-9\).

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -1/3x + 11\)

Step 1: Find the slope of the given line

To find the slope of the line with an x-intercept of 3 and a y-intercept of -9, we can use the formula \(m= (-9 - 0)/(0 - 3) = 3\)

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original slope, so the slope of the line we're looking to define would be \(m = -1 / 3\)

Step 3: Use the point-slope form of the equation

We can use the known point (-5,6) and the slope to get the y-intercept, the forms looks like this: \(Y - Y1 = m(X - X1)\), by substituting we get \(y - 6 = -1/3(x + 5)\)

Step 4: Convert to slope-intercept form

The slope-intercept form of a line's equation is \(y = mx + b\). By solving for \(y\) we get \(y = -1/3x + 5 + 6\)

Step 5: Simplify the equation

After simplification we get \(y = -1/3x + 11\).

Key Concepts

Perpendicular Lines

Perpendicular lines are an essential concept in geometry. They meet at a right angle, which is 90 degrees. When analyzing the slope of perpendicular lines, it's important to note that the product of their slopes is always -1.
If you have two lines and you know the slope of one, you can easily find the slope of the second (perpendicular) line. Simply take the negative reciprocal of the first line's slope.

• If the slope of the first line is \( m \), then the slope of the perpendicular line is \( -\frac{1}{m} \).
• This means if you have a line with slope 3, the perpendicular line will have a slope of \( -\frac{1}{3} \).

This relationship helps in finding equations of lines in various forms, including slope-intercept form.

Point-Slope Form

The point-slope form of a linear equation is a wonderful tool for writing equations of lines.It uses a point on the line and its slope to define the equation.
The point-slope form is expressed as \( y - y_1 = m (x - x_1) \), where \( (x_1, y_1) \) is a specific point on the line and \( m \) is the slope.
This form is particularly useful when you know:

• A point through which the line passes.
• The slope of the line.

When given a point and the slope, you can quickly write the equation of the line that passes through that point with the specified slope. From there, it’s easy to convert to other forms, such as slope-intercept form, by simply solving for \( y \).

Linear Equation

Linear equations are equations of the first degree, meaning they have no exponents greater than one.They produce straight lines when graphed on a coordinate plane.
The standard form of a linear equation is \( Ax + By = C \), but linear equations are often written in different forms depending on the information given:

• Slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• Point-slope form: Uses a known point and the slope to form the equation (as previously discussed).

Learning how to transition a linear equation between different forms is a key skill in algebra.Each form provides unique insights into the characteristics of the line, such as its slope, intercepts, and the shape it takes on a graph.