Problem 16 - Elementary Algebra

Question

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form. slope \(=\frac{2}{3}, \quad\) passes through (-1,2)

Step-by-Step Solution

Answer

Answer: The equation of the line is \(y=\frac{2}{3}x+\frac{8}{3}\).

Step 1: Use point-slope formula

The point-slope formula is given as follows: \((y - y_{1})=m(x - x_{1})\). Substitute the slope (\(\frac{2}{3}\)) and the point (-1, 2) into the formula: \((y-2)=\frac{2}{3}(x+1)\)

Step 2: Distribute the slope

Multiply \(\frac{2}{3}\) by both terms in the parentheses to distribute the slope: \(y-2=\frac{2}{3}x+\frac{2}{3}\)

Step 3: Move the 2 to the right side

Add 2 to both sides of the equation to isolate 'y': \(y=\frac{2}{3}x+\frac{2}{3}+2\)

Step 4: Simplify the right side

Write 2 as a fraction with the same denominator (3): \(y=\frac{2}{3}x+\frac{2}{3}+\frac{6}{3}\) Now, add the fractions together: \(y=\frac{2}{3}x+\frac{8}{3}\)

Step 5: Final Answer

The equation of the line in slope-intercept form is: \(y=\frac{2}{3}x+\frac{8}{3}\)

Key Concepts

Point-Slope Formula

Understanding the point-slope formula is crucial for anyone diving into the world of linear equations. It provides a direct way to create the equation of a line when you're given a point it passes through, and its slope.

Imagine drawing a line on a graph without any tools—just a point and an inclination of how steep the line should go. That's what the point-slope formula allows you to do. This formula is \((y - y_{1}) = m(x - x_{1})\), where \(m\) represents the slope, while \((x_1, y_1)\) represents the coordinates of the given point. By plugging in the values, you lay down the backbone of your linear equation.

Distributive Property

The distributive property is your algebraic sidekick, simplifying expressions and making it easier to solve equations. It's expressed as \(a(b + c) = ab + ac\). In essence, it states that if you have a number or a variable outside the parentheses, it needs to 'get acquainted' with each term inside.

If used correctly, the distributive property not only makes calculations neater but ensures accuracy when dealing with more complex algebraic expressions. When applying it to the point-slope formula, as seen in the exercise, we multiply the slope, a fraction, across each term in the parentheses, thus staying in sync with the property's 'distribute, then solve' philosophy.

Linear Equations

A linear equation is like a universal language in mathematics, spanning various levels of complexity. It's an equation that makes a straight line when graphed, hence 'linear'.

These equations typically look something like \(y = mx + b\), which is known as the slope-intercept form. Here, \(m\) denotes the slope of the line (how tilted it is), and \(b\) is the y-intercept (where the line crosses the y-axis). The elegance of linear equations lies in their simplicity and the wealth of information they provide about a line's behavior with just two parameters: slope and intercept.

To get an equation into this form, you'll often perform operations like the distributive property, as you did in our example problem, and isolate the variable 'y' to make the relationship between 'x' and 'y' crystal clear.