Problem 34 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ y=x $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 1, and the y-intercept is 0.

Step 1: Identify the coefficients and constant term

In the given equation, \(y=x\), the coefficients for x and y are both 1 (since there is no number written in front of x, we assume it to be 1). The constant term is also 0 because there is no additional constant.

Step 2: Determine the slope

The slope, \(m\), of the line can be found by looking at the coefficient of the x term. In this case, the coefficient of x is 1. So the slope of the line is 1.

Step 3: Determine the y-intercept

The y-intercept, \(b\), can be found by looking at the constant term. In this case, the constant term is 0. So the y-intercept of the line is also 0.

Step 4: Write the final answer

The slope of the line, \(m\), is 1 and the y-intercept, \(b\), is 0.

Key Concepts

Linear Equations

A linear equation is an algebraic expression that forms a straight line when plotted on a graph. It is typically written in the form:

• \( y = mx + b \)

where \( m \) is the slope, and \( b \) is the y-intercept. Linear equations are essential for understanding relationships where change occurs at a constant rate. For example, in the equation \( y = x \), the slope \( m \) is 1, meaning for every unit increase in \( x \), \( y \) increases by the same amount. The y-intercept \( b \) here is 0, indicating that the line crosses the y-axis at the origin. This simplicity makes linear equations a fundamental tool in mathematics as well as in real-life applications, such as calculating speed, distance, or cost.

Algebra

Algebra is the branch of mathematics that deals with expressions, equations, and variables. It involves solving equations to find unknown values. In the example \( y = x \), algebraically, we determine the slope by recognizing the coefficient of \( x \) as 1. This represents the steepness of the line. The y-intercept is found by identifying the constant term, which, in this equation, is 0. Algebra also introduces operations like adding, subtracting, multiplying, and dividing to manipulate equations and solve for unknowns. Mastering these skills is critical for tackling more complex problems in mathematics and other disciplines. Algebra helps us translate real-world problems into mathematical ones, making complex ideas easier to manage and analyze.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry. It focuses on defining and analyzing geometric shapes using a coordinate system. By plotting the equation \( y = x \) on a coordinate plane, we get a straight line with a slope of 1 passing through the origin (0,0). This line shows how \( x \) and \( y \) change together, providing a visual understanding of the relationship described by the equation. By using coordinates, we can determine important features like:

• Slope: Ratio of the vertical change to the horizontal change.
• Y-intercept: Point where the line crosses the y-axis.
• Distance and midpoint between points.

Coordinate geometry makes visualizing and solving problems involving shapes, graphs, and lines much simpler. It is a key concept for fields like engineering, physics, and computer graphics.