Problem 42 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ \left(\frac{1}{4}, 2\right),\left(-5, \frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The final equation of the line that passes through the points (1/4, 2) and (-5, 2/3) can be obtained by substituting the calculated slope 'm' and y-intercept 'b' into the slope-intercept form \( y = m*x + b \).

Step 1: Calculate the Slope

The slope 'm' of a line that passes through two points (x1, y1) and (x2, y2) is given by: \( m = \frac{y2 - y1}{x2 - x1} \) Here, (x1, y1) is (1/4, 2) and (x2, y2) is (-5, 2/3). Substituting these values into the formula gives: \( m = \frac{2/3 - 2}{-5 - 1/4} \) Simplifying this expression will give the calculated slope.

Step 2: Calculate the Y-Intercept

Once the slope 'm' is known, it can be substituted back into the slope-intercept equation, along with the coordinates of one of the points, to find the y-intercept 'b'. The equation is given by: \( b = y1 - m*x1 \) Substitute y1 = 2, x1 = 1/4 and 'm' as the calculated slope into the equation, and simplify the expression to find the value of 'b'.

Step 3: Write the Equation of the Line

After calculating the slope 'm' and the y-intercept 'b', substitute these values into the slope-intercept form. The final equation of the line will be: \( y = m*x + b \)

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is important when working with linear equations. The slope tells us how steep the line is and in which direction it tilts. It can be positive, negative, zero, or undefined. To find the slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

• If the slope is positive, as the \(x\) values increase, the \(y\) values rise, making the line go upward.
• If the slope is negative, as \(x\) values increase, the \(y\) values decrease, indicating a downward trend.
• A slope of zero means the line is perfectly horizontal and does not rise or fall.
• An undefined slope, often shown with a vertical line, means the \(x\) coordinates are the same, leading to a division by zero.

It's crucial to perform these calculations accurately to ensure you have the correct slope for your line, which directly affects the linear equation you derive from it.

Y-Intercept

In a linear equation, the y-intercept is the exact location where the line crosses the y-axis. This point is crucial because it gives you a starting point for graphing the line on a coordinate plane. The general form for finding the y-intercept from the slope-intercept form, \(y = mx + b\), is to identify \(b\) which is the constant term.
In cases where you have calculated the slope and have one point, \((x_1, y_1)\), you can find the y-intercept using: \[ b = y_1 - m \cdot x_1 \].
Once you calculate \(b\), plug it back into the slope-intercept form to finalize your linear equation.

• The y-intercept is often easy to identify when you have data points on a graph.
• It is the value of \(y\) when \(x\) is zero.

Knowing the y-intercept allows you to easily draw the line by starting from this specific point on the y-axis.

Linear Equations

Linear equations are foundational in algebra, facilitating the study of relationships between variables. A standard form of the linear equation is \(y = mx + b\), known as the slope-intercept form, where \(m\) represents the slope and \(b\) signifies the y-intercept. These equations form straight lines on a graph, displaying a consistent rate of change.
Understanding linear equations involves recognizing that:

• The slope \(m\) determines the line's direction and angle.
• The y-intercept \(b\) pins down the line's starting point on the y-axis.
• These concepts allow us to model and solve real-world problems, predicting values and trends.

Developing a strong grasp of linear equations helps you find solutions to problems involving relationships between two variables, making it an essential skill in both academic and real-life contexts.