Problem 64
Question
State (by writing it) the relationship being expressed by the equation \(y=3 x+5\).
Step-by-Step Solution
Verified Answer
Answer: In the linear equation \(y = 3x + 5\), the slope (3) represents a positive relationship between \(x\) and \(y\), with \(y\) increasing by 3 units for every unit increase in \(x\). The y-intercept (5) represents the value of \(y\) when \(x = 0\), indicating the point (0, 5) where the line intersects the y-axis.
1Step 1: Identify the Slope and Y-intercept
The given equation is \(y = 3x + 5\). In the slope-intercept form, which is \(y = mx + b\), the coefficient of \(x\) (m) is the slope and the constant term (b) is the y-intercept. In our equation, the slope (\(m\)) is 3 and the y-intercept (\(b\)) is 5.
2Step 2: Analyze the Slope
The slope represents the rate of change of \(y\) with respect to \(x\). In our equation, the slope is 3, which means that for every unit increase in \(x\), \(y\) will increase by 3 units. This indicates a positive linear relationship between \(x\) and \(y\). If the slope were negative, it would indicate a negative relationship.
3Step 3: Analyze the Y-intercept
The y-intercept represents the value of \(y\) when \(x\) is 0. In our equation, the y-intercept is 5, which means that when \(x = 0\), \(y = 5\). This point (0,5) is where the line representing the relationship between \(x\) and \(y\) intersects the y-axis.
4Step 4: Conclusion
The given equation \(y = 3x + 5\) expresses a linear relationship between \(x\) and \(y\). The slope (3) represents a positive relationship, where \(y\) increases by 3 units for every unit increase in \(x\). The y-intercept (5) indicates that when \(x = 0\), \(y = 5\).
Key Concepts
Slope-Intercept FormRate of ChangeY-InterceptPositive Linear Relationship
Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is essential for analyzing and graphing straight lines. This form is expressed as \( y = mx + b \), where \( m \) stands for the slope, and \( b \) represents the y-intercept. It provides a straightforward way to identify how a line behaves. With \( y = 3x + 5 \), the slope is 3 and the y-intercept is 5, making the equation easy to interpret and graph.
The slope-intercept form is useful because it immediately gives us two significant pieces of information about the line: how steep the slope is and where it crosses the y-axis. This allows us to quickly sketch the line without extensive calculations and provides a foundation for understanding more complex algebraic concepts.
The slope-intercept form is useful because it immediately gives us two significant pieces of information about the line: how steep the slope is and where it crosses the y-axis. This allows us to quickly sketch the line without extensive calculations and provides a foundation for understanding more complex algebraic concepts.
Rate of Change
In mathematics, the rate of change is an essential concept, especially when dealing with linear equations. It measures how one variable changes in relation to another. The slope of the linear equation \( y = mx + b \) is the rate of change, which indicates how much the variable \( y \) changes for a unit change in the variable \( x \).
A positive slope, like the 3 in our equation \( y = 3x + 5 \), shows a direct relationship - as \( x \) increases, so does \( y \). This can be visualized on a graph, where each step to the right along the x-axis results in a step upward along the y-axis. Conversely, a negative slope would mean that as \( x \) increases, \( y \) decreases. Understanding the slope as a rate of change helps us to interpret real-world situations, like speed, price cost, or temperature changes over time.
A positive slope, like the 3 in our equation \( y = 3x + 5 \), shows a direct relationship - as \( x \) increases, so does \( y \). This can be visualized on a graph, where each step to the right along the x-axis results in a step upward along the y-axis. Conversely, a negative slope would mean that as \( x \) increases, \( y \) decreases. Understanding the slope as a rate of change helps us to interpret real-world situations, like speed, price cost, or temperature changes over time.
Y-Intercept
The y-intercept is a key component of the slope-intercept form of a linear equation. It is the point where the line crosses the y-axis on a graph. Represented by \( b \) in the equation \( y = mx + b \), it marks the value of \( y \) when \( x \) is zero. In the context of our example, the y-intercept is 5, denoted by the point (0,5) on the graph.
It’s important because it can represent the starting value in many real-life situations. For instance, if you're looking at a graph that shows how much money you have over time, the y-intercept would show your initial amount before any changes occur. It gives us a clue about where to begin plotting our line on a graph and helps us envision where the variables in our equation started off before any change took place.
It’s important because it can represent the starting value in many real-life situations. For instance, if you're looking at a graph that shows how much money you have over time, the y-intercept would show your initial amount before any changes occur. It gives us a clue about where to begin plotting our line on a graph and helps us envision where the variables in our equation started off before any change took place.
Positive Linear Relationship
A positive linear relationship is one where as one variable increases, the other variable also increases at a constant rate. This kind of relationship is graphically represented by a line with an upward slope. In our example equation \( y = 3x + 5 \), the relationship between \( x \) and \( y \) is positive because for every increase in \( x \), there is a proportional increase in \( y \), as indicated by the positive slope which is 3. This tells us that for every step we move horizontally to the right (an increase in \( x \)), we move three steps vertically up (an increase in \( y \)).
This concept is crucial because it exists widely in our everyday experiences, such as the correlation between hours worked and money earned or the connection between the amount of study time and the scores achieved on a test. Recognizing this pattern allows for making predictions and understanding the consistency of the relationship between variables.
This concept is crucial because it exists widely in our everyday experiences, such as the correlation between hours worked and money earned or the connection between the amount of study time and the scores achieved on a test. Recognizing this pattern allows for making predictions and understanding the consistency of the relationship between variables.
Other exercises in this chapter
Problem 63
For the following problems, find the products. $$ \left(2 b+\frac{6}{7}\right)\left(2 b-\frac{6}{7}\right) $$
View solution Problem 64
For the following problems, simplify each of the algebraic expressions. $$ 1\left(2 x-6 b+6 a^{2} b+8 b^{2}\right)+1\left(5 x+2 b-3 a^{2} b\right) $$
View solution Problem 64
For the following problems, perform the multiplications and combine any like terms. $$ 2 x^{4}\left(6 x^{3}-5 x^{2}-2 x+3\right) $$
View solution Problem 64
For the following problems, note how many: $$ (x+1) \text { 's in } 8(x+1) ? $$
View solution