Problem 16 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-5,\) passing through \((-4,-2)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 2 = -5(x + 4)\) and in slope-intercept form is \(y = -5x - 22\)

Step 1: Apply Point-Slope Form

Start by plugging the given values into the point-slope form. Here \(m=-5,\) \(x_1=-4,\) and \(y_1=-2\). The equation, after substitution, will be: \(y - (-2) = -5(x - (-4))\)

Step 2: Simplify the Point-Slope Equation

Now simplify the equation from step 1. The equation becomes \(y + 2 = -5(x + 4)\). After simplifying this further, you get \(y = -5x - 20 - 2\). Further simplification yields: \(y = -5x - 22\)

Step 3: Write the Equation in Slope-Intercept Form

The equation of the line after simplification is already in slope-intercept form, which is \(y = mx + b\). In this case, \(m = -5\), and \(b = -22\)

Key Concepts

Slope-Intercept Form

The equation of a line in the slope-intercept form is one of the most common and useful for students to know. It is given by the formula \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, which is where the line crosses the y-axis.

This form is particularly useful because it allows for quick identification of the slope and y-intercept from a glance. Knowing these components can help in graphing the equation or understanding how changes in them transform the line visually. For instance:

• A positive slope \(m\) means the line rises as it moves from left to right.
• A negative slope \(m\) indicates the line falls from left to right.
• The y-intercept \(b\) provides a starting point on the graph where the line will cross the y-axis.

Linear Equations

Linear equations are equations that form a straight line when graphed on a coordinate plane. These equations are typically denoted in the form \(ax + by = c\) or more commonly in slope-intercept form \(y = mx + b\).

The fundamental characteristic of linear equations is that their solutions form a straight, unending line across the graph. This is because they represent relationships where the rate of change between the two variables \(x\) and \(y\) is constant. This constant rate of change is defined by the slope, \(m\).

Linear equations arise in countless real-world situations, including calculating distances, predicting costs, and converting temperatures. When solving linear equations in real-life scenarios or simplifying them in algebra, remember these key steps:

• Identify known variables and constants.
• Arrange the equation into a recognizable form, such as the slope-intercept form.
• Solve for the desired variable using algebraic principles like distributing and combining like terms.

Algebra Problems

Algebra problems often challenge students to apply rules and concepts to find unknown values. One type of common problem involves finding the equation of a line given certain information like a point and a slope.

The process often starts with using the point-slope form \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. By substituting these known values into the formula, it provides a starting equation that can be manipulated.

Once the point-slope form is established, the equation can be simplified into slope-intercept form, facilitating easier graphing and understanding. Key steps to consider when tackling these problems include:

• Accurately identify and substitute values into the initial equation.
• Carefully simplify by distributing the slope \(m\) across terms.
• Rearrange terms to solve for \(y\) in terms of \(x\), achieving the slope-intercept form \(y = mx + b\).

These algebra problems hone skills in making connections between different forms of equations and developing critical thinking to arrive at solutions.