Problem 19 - Algebra 1: Concepts and Skills

Question

Find the slope and y-intercept of the graph of the equation. $$y=2 x-9$$

Step-by-Step Solution

Answer

The slope of the line is 2 and the y-intercept is -9.

Step 1: Identify the Slope

In the given equation \(y=2x-9\), \(m=2\) is the coefficient of \(x\), so the slope of the line is 2.

Step 2: Identify the Y-intercept

In the given equation \(y=2x-9\), \(-9\) is the constant term, thus it is the y-intercept. This means the line will cross the y-axis at the point \((0, -9)\).

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations look like this: \(y = mx + b\), where \(m\) and \(b\) are constants. The constant \(m\) represents the slope of the line, which dictates the angle and direction of the line on a graph. The constant \(b\) is the y-intercept, which tells us where the line crosses the y-axis.

In the context of our given exercise \(y = 2x - 9\), the equation is in standard linear form, with the slope \(m\) being 2 and the y-intercept \(b\) being -9. The significance of understanding linear equations stems from their pervasiveness in various real-world scenarios, such as calculating expenses over time, determining speed, or even modeling population growth under constant growth rates.

Graphing Linear Equations

Graphing linear equations involves plotting the relationship between two variables on a coordinate plane to form a straight line. The most efficient way to graph a linear equation is by using the slope and y-intercept.

For the equation \(y = 2x - 9\), you begin graphing by pinpointing the y-intercept on the graph, which is the point where the line crosses the y-axis. Here, our y-intercept is \((0, -9)\). After marking this point on the y-axis, you use the slope to determine the direction and steepness of the line. The slope \(m = 2\) means that for every unit increase in \(x\), \(y\) increases by 2 units. So, starting from the y-intercept, you would move right 1 unit (positive direction along the x-axis) and up 2 units (positive direction along the y-axis) to mark another point. Drawing a line through both points will give you the graph of the equation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. For example, in the expression \(2x - 9\), \(2x\) is a term that represents the multiplication of the variable \(x\) with the constant 2, and \(-9\) is a term representing a negative constant. The entire expression forms a part of our linear equation \(y = 2x - 9\).

Algebraic expressions are the building blocks of algebraic equations, and understanding them is fundamental to solving mathematical problems. They do not have an 'equals' sign like equations do, but when placed into an equation such as \(y = 2x - 9\), they can define the relationship between variables that we can then graph. When visualizing algebraic expressions, it's important to conceptualize how changing the value of \(x\), for instance, will affect the value of \(y\) when the expression is a part of an equation.