Problem 15 - Technical Mathematics with Calculus

Question

Write the equation of each line in slope-intercept form. Slope \(=2 ;\) passes through (3,4)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is y = 2x - 2.

Step 1: Identify the slope-intercept form

The slope-intercept form of a linear equation is given by the formula y = mx + b, where m is the slope and b is the y-intercept, the point where the line crosses the y-axis.

Step 2: Use the given slope

According to the exercise, the slope (m) is 2. Substitute m = 2 into the slope-intercept equation, which gives y = 2x + b.

Step 3: Use the given point to find the y-intercept (b)

The line passes through the point (3, 4), which means when x = 3, y = 4. Substitute these values into the equation to find b: 4 = 2(3) + b.

Step 4: Solve for the y-intercept (b)

Solving the equation from Step 3, 4 = 6 + b, we find b = 4 - 6 = -2.

Step 5: Write the final equation

Substitute b = -2 into the equation from Step 2, which gives the final equation of the line: y = 2x - 2.

Key Concepts

Understanding Linear Equations

Linear equations are the simplest form of equations you'll encounter in algebra. They're called 'linear' because their solutions graph to a straight line. The general form of a linear equation in two variables (typically x and y) can be expressed as ax + by = c, where a, b, and c are constants. What makes the slope-intercept form of a linear equation special is its utility in quickly sketching a line when provided with the slope and y-intercept values.

In the context of the given exercise, knowing how to translate the slope and a point on the line into the slope-intercept equation is key for not just finding the equation of a line, but also understanding the relationship between the algebraic representation of a line and its graphical representation.

The Slope-Intercept Form Equation

The slope-intercept form is incredibly user-friendly because it directly gives you the slope and the y-intercept of a line. It's written as \( y = mx + b \), where \( m \) is the slope, and \( b \) indicates the y-intercept — the point where the line crosses the y-axis. This form makes it easy to graph linear equations by simply plotting the y-intercept and then using the slope to find another point.

When given a slope and a specific point on a line like in our exercise, the slope-interfect form allows us to plug these values in to find the unknown y-intercept. This can be particularly helpful in situations where you need to visualize data trends or quickly sketch graphs without the need for plotting multiple points.

Calculating the Y-intercept

Calculating the y-intercept (\b) when given a slope and a point is a straightforward process. With the slope-intercept form (\( y = mx + b \)), the slope (\b) is given, and you just need one point that lies on the line to solve for the y-intercept. By inserting the x and y values of the point into the equation, you create a simple equation that can be solved for \( b \). This value of \( b \) tells you exactly where the line will cross the y-axis if you were to graph it, providing a starting point for the line and a clear vision of how it will tilt upwards or downwards depending on the sign of the slope.

The calculated y-intercept is also essential for understanding the starting value of a variable in applied problems, such as predicting profits or population sizes at the starting time of a study. The ability to calculate the y-intercept is an invaluable tool for translating real-world situations into mathematical models.