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Chapter 13: Problem 24

Find the slope and \(y\) -intercept of each line. $$4 x+3 y=8$$

Short Answer

Expert verified
The slope of the line is \(-\frac{4}{3}\) and the y-intercept is \(\frac{8}{3}\).

Step by step solution

01

Rewrite the equation in slope-intercept form

The first step is to arrange the given equation \(4x + 3y = 8\) in the form of \(y = mx + b\). This can be achieved by subtracting \(4x\) from both sides of the equation, to isolate \(y\). This gives \(3y = -4x + 8\). We then divide the entire equation by 3 to solve for \(y\), which results in \(y = -\frac{4}{3}x + \frac{8}{3}\).
02

Identify the slope and y-intercept

Once the equation is in the form \(y = mx + b\), it's easy to identify the slope \(m\) and the \(y\)-intercept \(b\). From the equation \(y = -\frac{4}{3}x + \frac{8}{3}\), it can be seen that the slope \(m\) is \(-\frac{4}{3}\) and the \(y\)-intercept \(b\) is \(\frac{8}{3}\).

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Key Concepts

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

Linear Equations
A linear equation is like a line drawn on a sheet of graph paper. It's an equation that forms a straight line when it's plotted on a graph. These equations typically have two variables, often represented by \(x\) (the independent variable) and \(y\) (the dependent variable). The general form of a linear equation is usually written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constant numbers. The key feature of a linear equation is that the highest power of the variable is one, making it a linear expression.

  • Linear equations are simple yet powerful, modeling relationships with a straight-line graph.
  • They help in finding values where two quantities are directly proportional, meaning they increase or decrease in constant rates.
Understanding linear equations opens the door to more complex algebraic concepts as they play a foundational role in different mathematical practices.
Slope
The slope of a line is a measure of how steep the line is. In the equation \(y = mx + b\), the slope is represented by the letter \(m\). The slope can be thought of as "rise over run," which describes how much \(y\) changes as \(x\) changes. Mathematically, it's expressed as: \[ m = \frac{\text{change in } y}{\text{change in } x} \] An easier way to understand is: if you move one unit to the right along the \(x\)-axis, the slope tells you how many units you go up or down the \(y\)-axis.

  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal and \(y\) doesn't change as \(x\) changes.
Slope is crucial because it describes the rate and direction of change between variables in a linear relationship.
Y-Intercept
The \(y\)-intercept is where the line crosses the \(y\)-axis in a graph. In the slope-intercept form \(y = mx + b\), the \(y\)-intercept is indicated by the letter \(b\). This point on the graph is crucial as it serves as the starting point of the line when \(x = 0\). Here's how to visualize it:

  • If you imagine the graph as a map, the \(y\)-intercept tells you where you begin your journey onto the line.
  • A positive \(y\)-intercept means the line crosses above the origin, and a negative \(y\)-intercept crosses below the origin.
The \(y\)-intercept helps in fully defining the line along with slope, enabling us to sketch the line accurately on a graph. By knowing \(b\), you're better equipped to understand the position and direction of the given line.

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Most popular questions from this chapter

Decide what special type of quadrilateral \(H I J K\) is. Then prove that your answer is correct. \(H(7,5)\) \(I(8,3)\) \(J(0,-1)\) \(K(-1,1)\)

line through \((8,0)\) and perpendicular to the line \(3 x+4 y=12\)

Find the center of the circle that passes through \((2,10),(10,6),\) and \((-6,-6)\)

line through \((-1,3)\) and parallel to the line \(3 x+5 y=15\)

Find and then compare lengths of segments. Quadrilateral \(T A U L\) has vertices \(T(4,6), A(6,-4), U(-4,-2),\) and \(L(-2,4) .\) Show that the diagonals are congruent.
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