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Chapter 7: Problem 66

Describe how to find the \(y\)-intercept of a linear equation.

Short Answer

Expert verified
The \(y\)-intercept of a linear equation can be found by transforming the given equation into slope-intercept form, \(y = mx + b\). The \(y\)-intercept is the \(b\) term in this equation, representing the point where the line crosses the y-axis.

Step by step solution

01

Identification

Identify the structure of your linear equation. If your equation is not in the form \(y = mx + b\) or \(Ax + By = C\), you will first need to transform it into one of these forms.
02

Transformation into Slope-Intercept Form

If your equation is in the form \(Ax + By = C\), we want to transform it into slope-intercept form. Do this by isolating \(y\) on one side. Subtract \(Ax\) from both sides, which then gives us \(By = -Ax + C\). Now, divide through by \(B\) to get \(y = -A/Bx + C/B\). Now, it's in the form \(y = mx + b\).
03

Obtaining the Y-Intercept

Now that we have the equation in the form \(y = mx + b\), we can identify the \(y\)-intercept. The \(y\)-intercept is simply the \(b\) term in the equation, because this is the value of \(y\) when \(x = 0\). This means, the line intercepts the y-axis at \(b\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Equations
A linear equation is a type of equation that represents a straight line when graphed on a coordinate plane. It typically takes the form of \(y = mx + b\) or \(Ax + By = C\). In this context, \(x\) and \(y\) are variables, while \(m\), \(b\), \(A\), \(B\), and \(C\) are constants.

The characteristic feature of linear equations is that they have no exponents higher than one, which means they form a straight line when plotted. Here are some key points about linear equations:
  • Their graphs are straight lines.
  • The "m" in the equation \(y = mx + b\) represents the slope of the line.
  • The "b" represents the y-intercept.
Understanding these components is crucial for graphing linear equations, finding the slope, and identifying where the line intersects the y-axis.
The Role of Slope-Intercept Form in Linear Equations
The slope-intercept form of a linear equation is \(y = mx + b\). This particular form is one of the most common ways to express a linear equation because it allows for easy identification of two key components: the slope and the y-intercept.

Here's how it works:
  • "m" is the slope of the line, which tells us how steep the line is. A positive \(m\) means the line slopes upwards, while a negative \(m\) indicates it slopes downwards.
  • "b" is the y-intercept, which is the point where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero.
This form is extremely useful for graphing because:
  • It provides a direct visualization of the slope and y-intercept, helping in plotting the line on a graph.
  • It simplifies the process of solving problems related to changes in the line, such as shifts or rotations.
Converting equations into the slope-intercept form can thus simplify many algebraic problems involving linear equations.
Coordinate Geometry and the Y-intercept
Coordinate geometry involves plotting points, lines, and curves on an x-y plane. In this space, the y-intercept is a critical intersection point to understand. It represents the point where a line intersects the y-axis. When working with the y-intercept in coordinate geometry, keep these points in mind:

  • The y-axis represents all points where \(x = 0\).
  • Finding the y-intercept involves determining the y-coordinate when \(x = 0\). In the equation \(y = mx + b\), this value is precisely "b".
  • The y-intercept provides a starting point on the graph, making it easier to then draw the line using the slope \(m\).
Effectively, the y-intercept helps to navigate and interpret the position of linear equations on a coordinate plane. It's instrumental for understanding how different lines behave and how they relate to one another in a geometric setting.

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Most popular questions from this chapter

The value of \(a\) in \(y=a x^{2}+b x+c\) and the vertex of the parabola are given. How many \(x\)-intercepts does the parabola have? Explain how you arrived at this number. \(a=1 ;\) vertex at \((2,0)\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \leq 3 \\ y>-1\end{array}\right.\)

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { x, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y} \text {, Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 1 & 7.6 \\ \hline 3 & 6 \\ \hline 4 & 2.8 \\ \hline \end{array} $$ QuadReg $$ \begin{aligned} &y=a x^{2}+b x+c \\ &a=-.8 \\ &b=2.4 \\ &c=6 \end{aligned} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs feet from where it was thrown and the maximum height is feet.

The data can be modeled by $$ f(x)=956 x+3176 \text { and } g(x)=3904 e^{0.134 x} \text {, } $$ in which \(f(x)\) and \(g(x)\) represent the average cost of room and board at public four-year colleges in the school year ending \(x\) years after 2010. Use these functions to solve Exercises 33-34. Where necessary, round answers to the nearest whole dollar. a. According to the linear model, what was the average cost of room and board at public four-year colleges for the school year ending in 2015 ? b. According to the exponential model, what was the average cost of room and board at public four-year colleges for the school year ending in 2015 ? c. Which function is a better model for the data for the school year ending in \(2015 ?\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 2 \\ y \leq 3\end{array}\right.\)
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