Problem 10 - 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 \(=-1,\) passing through \(\left(-4,-\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -x - \frac{17}{4}\).

Step 1: Insert the slope and point into the point-slope form

Write the point-slope form of the equation of the line using the slope \(-1\) and the point \(-4, -1/4\). The result is \(y - (-\frac{1}{4}) = -1 (x - (-4))\)

Step 2: Simplify the equation

Simplify the equation by distributing \(x\) and simplifying the numbers. That gives us \(y + \frac{1}{4} = -x - 4\)

Step 3: Transform the equation into slope-intercept form

To convert to slope-intercept form, isolate \(y\) on one side. Subtract \(\frac{1}{4}\) from both sides to get \(y = -x - 4 - \frac{1}{4}\). Then, simplify to \(y = -x - \frac{17}{4}\)

Key Concepts

Slope-Intercept Form

When diving into linear equations, the slope-intercept form is a popular topic. It represents a line on the graph with utmost simplicity and clarity. The form can be given as:
\[ y = mx + b \]
where, \( m \) denotes the slope of the line, and \( b \) corresponds to the y-intercept, which is where the line crosses the y-axis.

• The \( m \) value tells us how steep the line is or, in other words, how much \( y \) changes for a unit change in \( x \) (rise over run).
• The \( b \) value indicates the precise point on the y-axis which the line crosses.

This form is particularly useful to quickly sketch a graph or to determine at a glance how lines will intersect or run parallel.

Linear Equations

Linear equations are fundamental in algebra and are used to describe relationships between two variables that create a straight line when plotted on a graph.

A general linear equation in two variables, \( x \) and \( y \) has the form:
\[ ax + by = c \]
where \( a \) and \( b \) are coefficients, and \( c \) is a constant. What’s special about these equations is that they always produce a graph with a uniform rate of increase or decrease, hence ‘linear’. They are invaluable for modeling situations with constant growth or decay, solving problems with multiple factors, and finding relationships within data points.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

Unlike equations, expressions don't have an equal sign; they're not statements of equality but rather mathematical phrases that can be simplified or factored.

Parts of An Algebraic Expression

• Variable: A symbol representing a quantity that can vary, usually \( x \) or \( y \).
• Coefficient: The numerical factor of a term with a variable.
• Constant: A fixed value that doesn’t change.

We use these expressions to describe patterns, find unknowns, and express the general rules of operations.

Simplifying Equations

Simplifying equations is a key skill in algebra, which reduces equations to their simplest form, making them easier to understand and solve. The process usually involves combining like terms, distributing through parentheses, and isolating the variables.

It's like cleaning up a messy room; by putting everything where it belongs, the layout becomes clear, and you can see the floor again - or in this case, the solution to the equation.

Steps in Simplifying

• Combine like terms: terms with the same variable and exponent.
• Use the distributive property: \( a(b + c) = ab + ac \).
• Move terms: if you need to isolate a variable, add or subtract terms from both sides to achieve it.

Through these steps, you can transform an equation into a more workable version, often bringing you steps closer to finding the value of the variable.