Problem 17 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((1,7) ; \quad\) slope \(\frac{2}{3}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{2}{3}x + \frac{19}{3} \).

Step 1: Identify the Point-Slope Formula

The point-slope form of a line's equation is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.

Step 2: Substitute Given Values into the Formula

Using the point \((1, 7)\) and the slope \(\frac{2}{3}\), substitute these values into the point-slope formula. This gives us: \[ y - 7 = \frac{2}{3}(x - 1) \]

Step 3: Simplify the Equation

Distribute the slope \(\frac{2}{3}\) through the expression \( (x - 1) \): \[ y - 7 = \frac{2}{3}x - \frac{2}{3} \]

Step 4: Solve for y in Slope-Intercept Form

Add 7 to both sides of the equation to solve for \( y \): \[ y = \frac{2}{3}x - \frac{2}{3} + 7 \] Simplify the constant terms: \[ y = \frac{2}{3}x + \frac{19}{3} \]

Step 5: Final Equation

The equation of the line in slope-intercept form is \( y = \frac{2}{3}x + \frac{19}{3} \).

Key Concepts

Understanding Point-Slope Form

The point-slope form is a way to express the equation of a line using a slope and a specific point it passes through. It's given by the formula: \[ y - y_1 = m(x - x_1) \]where:

• \( m \) is the slope of the line, representing how steep the line is.
• \( (x_1, y_1) \) is a point on the line.

Anytime you know a line's slope and at least one point on the line, you can use the point-slope form to write its equation. For example, in the original exercise, we were given a slope of \( \frac{2}{3} \) and a point \( (1, 7) \). By substituting these values into the point-slope formula, we began shaping the line's equation. Always remember, this form is incredibly useful because it directly uses the slope and a specific point you know, offering an initial picture of the line.

Exploring Slope-Intercept Form

The slope-intercept form is another way to express a line's equation, often preferred for its simplicity and clarity. It's expressed as:\[ y = mx + b \]where:

• \( m \) is the slope.
• \( b \) is the y-intercept, where the line crosses the y-axis.

This form highlights how the y-value changes based on the x-value changes, and where the line sits on the y-axis when \( x = 0 \). To convert from point-slope to slope-intercept form, you solve the equation for \( y \). In our example, this process involved expanding the expression and moving the constant terms appropriately, resulting in \( y = \frac{2}{3}x + \frac{19}{3} \). Once in slope-intercept form, it's easy to graph or interpret the line visually because you can directly see the slope and where it hits the y-axis.

Fundamentals of Linear Equations

Linear equations are algebraic expressions that describe straight lines. The general form of a linear equation is usually simpler to work with either through the point-slope or slope-intercept forms. They allow you to predict how the line behaves and intersects two axes. Understanding linear equations is crucial because:

• They model constant rates of change, making them useful in real-world situations.
• They help analyze relationships between variables—for instance, predicting outcomes based on a set input.
• They form the basis for more complex algebraic concepts.

In simpler terms, a linear equation describes a set path. When we mention the concept as in our example, we are essentially defining this straight path with a formula that reflects constant growth or decline. Mastering these fundamentals is key to understanding more complex mathematics.