Problem 8 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=-11,\) passing through \((0,-3)\)

Step-by-Step Solution

Answer

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

Step 1: Write the equation in point-slope form

In this case, m (the slope) is -11, and the point (x1, y1) used is (0, -3). Following the general formula, the equation becomes \(y - (-3) = -11 (x - 0)\)

Step 2: Simplify the equation

The equation should be simplified now. First, subtract -3 from both sides of the equation to get rid of the negative sign before -3. This leads to \(y + 3 = -11x\)

Step 3: Rewriting in slope-intercept form

The equation should now be arranged in the form of \(y = mx + b\), which is also known as the slope-intercept form. Subtracting 3 from both sides of the equation, we get \(y = -11x - 3\) which is our final equation in slope-intercept form

Key Concepts

Slope-Intercept Form

The slope-intercept form is crucial for understanding and working with linear equations in algebra. It is represented by the equation \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. This form makes analyzing the rate of change and the initial value of a linear function straightforward. For instance, in our exercise, after simplifying the point-slope equation, we get \(y = -11x - 3\), which tells us directly the steepness of the line is -11, and the line crosses the y-axis at \(y = -3\).

By using the slope-intercept form, we can quickly sketch the graph of the line without additional calculations. If the slope is negative, as it is in our example, the line will tilt downwards from left to right, illustrating a decrease. As an improvement advice, remember that you can always check if the slope-intercept form of an equation was found correctly by plugging in the coordinates of a known point on the line and verifying if the equation holds true.

Linear Equations

At the heart of algebra, linear equations form the basis for much of the subject's problem-solving techniques. They represent relationships that have a constant rate of change, which graphically can be shown as straight lines. Linear equations can be written in various forms, including but not limited to the slope-intercept and point-slope forms.

Understanding how to convert between these forms is a valuable skill. As demonstrated in the exercise, we started with a point-slope form and manipulated it to obtain the slope-intercept form. It's essential to know that the equation of a line is the same regardless of the form—it's just a matter of which one is more useful for your current needs. To improve one's understanding, practice by converting linear equations between different forms and using them to model real-world situations.

Algebra

Algebra is much more than just solving for x. It's a language of patterns, an art of reasoning, and a tool that enables us to model and solve real-world problems. Learning to manipulate equations, like converting from point-slope to slope-intercept form, is one of algebra's fundamental skills. Here are a few tips to excel in algebra:

• Always perform the same operation on both sides of an equation.
• Understand the 'why'—don't just memorize formulas.
• Practice different types of problems frequently.

Our exercise exemplifies algebraic manipulation, showing how to handle negative signs and move terms from one side of an equation to the other. Ultimately, algebra is about finding patterns and using logical steps to solve for unknown variables. It's empowering—as you master algebra, you'll find that many complex problems in math and beyond become accessible and solvable.