Problem 9
Question
Find the slope and the y-intercept of the graph of the equation. $$ y=5 x-3 $$
Step-by-Step Solution
Verified Answer
The slope of the equation is 5 and the y-intercept is -3.
1Step 1: Identify the slope
By comparing the given equation \(y = 5x - 3\) with the slope-intercept form \(y = mx + c\), the coefficient of \(x\) in the equation is the slope. Hence, in this case, the slope (\(m\)) is 5.
2Step 2: Identify the y-intercept
From the equation \(y = 5x - 3\), the constant term is the y-intercept. The y-intercept (\(c\)) is thus -3.
Key Concepts
Linear equationsSlopeY-intercept
Linear equations
A linear equation is one of the first steps into the world of algebra. These equations graph as straight lines, and they follow a simple form called the slope-intercept form. This form is written as \(y = mx + c\). In this format, \(m\) is the slope of the line, and \(c\) is the y-intercept. Linear equations can model many real-world situations where changes occur at a constant rate.
Understanding linear equations can help in visualizing how two variables relate to each other. For example, in the equation \(y = 5x - 3\), each value of \(x\) affects \(y\) by altering it directly through a linear relationship. As the value of \(x\) changes, \(y\) increases by a proportion determined by the slope.
When solving linear equations, the goal is often to identify these two key components: the slope and the y-intercept.
Understanding linear equations can help in visualizing how two variables relate to each other. For example, in the equation \(y = 5x - 3\), each value of \(x\) affects \(y\) by altering it directly through a linear relationship. As the value of \(x\) changes, \(y\) increases by a proportion determined by the slope.
When solving linear equations, the goal is often to identify these two key components: the slope and the y-intercept.
Slope
The slope of a line is a measure of its steepness or tilt. In the slope-intercept form of a linear equation, the slope is represented by the letter \(m\). This value indicates how much \(y\) will change for a one-unit change in \(x\).
Think of the slope as "rise over run"—it describes how many units the line rises or falls vertically for each unit it moves horizontally. A positive slope like 5 in \(y = 5x - 3\) means the line rises as you move from left to right. Contrast this with a negative slope, where the line would fall as it moves across the graph.
Understanding slopes can help you determine how quickly things are changing. For instance, in economic models, the slope might tell you how much a product's supply increases as its price rises. By mastering the concept of slope, you can better understand and predict trends in data.
Think of the slope as "rise over run"—it describes how many units the line rises or falls vertically for each unit it moves horizontally. A positive slope like 5 in \(y = 5x - 3\) means the line rises as you move from left to right. Contrast this with a negative slope, where the line would fall as it moves across the graph.
Understanding slopes can help you determine how quickly things are changing. For instance, in economic models, the slope might tell you how much a product's supply increases as its price rises. By mastering the concept of slope, you can better understand and predict trends in data.
Y-intercept
The y-intercept is a crucial point where the line crosses the y-axis on a graph. It is represented by \(c\) in the slope-intercept form \(y = mx + c\). The y-intercept signifies the value of \(y\) when \(x\) is zero.
In our example equation \(y = 5x - 3\), the y-intercept is \(-3\). This means that when you plot the line on a graph, it will intersect the y-axis at the point (0, -3). This point is important because it gives a starting position from which the line can be drawn.
The y-intercept can be used in real-world scenarios such as finding the initial amount or starting point of growth or decay over time. By identifying the y-intercept, you gain insight into the underlying patterns present at the beginning of a sampling or observation period.
In our example equation \(y = 5x - 3\), the y-intercept is \(-3\). This means that when you plot the line on a graph, it will intersect the y-axis at the point (0, -3). This point is important because it gives a starting position from which the line can be drawn.
The y-intercept can be used in real-world scenarios such as finding the initial amount or starting point of growth or decay over time. By identifying the y-intercept, you gain insight into the underlying patterns present at the beginning of a sampling or observation period.
Other exercises in this chapter
Problem 9
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