Problem 28 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (1,4) \text { and }(-1,-4) $$

Step-by-Step Solution

Answer

The slope-intercept form of the line that passes through the points (1,4) and (-1,-4) is \( y = 4x \)

Step 1: Calculate the slope

First, calculate the slope(m) between the two points. The formula to calculate the slope is \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \). Plug the given points into this formula: \( m = \frac{(-4 - 4)}{(-1 - 1)} \)

Step 2: Simplify the slope

Simplify the resulting equation to get the slope: \( m = \frac{-8}{-2} = 4 \)

Step 3: Find the y-intercept

Now to calculate the y-intercept(c), plug in one of the points and the slope into the equation \( y = mx + c \). Choosing (1,4) for simplicity and rearranging the equation for c, we get \( c = y - mx = 4 - 4*1 = 0 \)

Step 4: Write the equation of the line

Finally, combine the computed y-intercept and slope to form the slope-intercept form of the linear equation. The equation is \( y = 4x + 0 \), which simplifies to \( y = 4x \)

Key Concepts

Linear Equations

Linear equations are fundamental in understanding the relationship between two variables, typically x and y, in algebra. Imagine you have a straight line on a graph; this line can be represented by a linear equation, which shows how y changes with respect to x. The most common form of a linear equation is the slope-intercept form, represented as \( y = mx + c \).

This equation tells you everything you need to know about the line:

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

Linear equations are vital because they are easy to work with. They can be applied in various real-world scenarios, such as predicting trends, analyzing data, or understanding relationships between variables.

Slope Calculation

The slope of a line in a graph is a measure of its steepness. Higher the value, the steeper the line and vice versa. Calculating the slope is an essential step in forming the slope-intercept equation.

To find the slope \( m \), we use the formula:

• \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \)

This formula measures how much the y-variable goes up or down as you move from one point to another on the x-axis. For example, using the points (1,4) and (-1,-4):

• \( m = \frac{(-4 - 4)}{(-1 - 1)} = \frac{-8}{-2} = 4 \)

Thus, the slope indicates the line rises by 4 units on the y-axis for every unit it moves right on the x-axis.

Y-Intercept

The y-intercept is a crucial part of the slope-intercept equation. It represents the point where the line crosses the y-axis. Knowing the y-intercept provides a fixed point through which the line passes, allowing you to graph it easily.

To find the y-intercept, once you have the slope, insert one of the given points into the line equation \( y = mx + c \) and rearrange to solve for \( c \).
For instance, using the point (1,4):

• \( c = y - mx = 4 - 4 \times 1 = 0 \)

In this example, the line crosses the y-axis at the origin (0,0), making the y-intercept \( c = 0 \). This helps in understanding the initial value of y when x is zero.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It's fundamentally about finding unknowns and solving equations. Linear equations, like \( y = mx + c \), are a key part of algebra, making it easier to understand relationships and predict outcomes based on given conditions.

In algebra, you often use algebraic expressions to form equations and solve for one or more unknown quantities. This process involves several steps:

• Identifying what you are asked to find.
• Setting up the equation(s) based on provided conditions.
• Simplifying the equations where necessary.
• Solving for the unknowns using algebraic techniques.

Algebra lays the groundwork for advanced mathematics and is a powerful tool for solving real-life problems. By mastering algebra, you can easily manipulate linear equations and comprehend complex mathematical ideas.