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Chapter 0: Problem 35

Write the equation in slope-intercept form. Identify the slope and the \(y\) -intercept. $$4 x-y=3$$

Short Answer

Expert verified
The slope is 4, and the y-intercept is -3.

Step by step solution

01

Understand the Equation Form

The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. Our goal is to rearrange the given equation into this form.
02

Isolate y on One Side

Start with the given equation: \[ 4x - y = 3 \] To isolate \(y\), subtract \(4x\) from both sides:\[ -y = -4x + 3 \]
03

Solve for Positive y

Currently, \(y\) is negative, so multiply every term in the equation by -1 to solve for \(y\):\[ y = 4x - 3 \]Now, the equation is in the slope-intercept form, \(y = mx + b\).
04

Identify the Slope and Y-Intercept

Comparing \(y = 4x - 3\) to the slope-intercept form \(y = mx + b\), identify:- Slope (\(m\)) is 4.- Y-intercept (\(b\)) is -3.

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Key Concepts

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

Linear Equation
A linear equation is a fundamental concept in algebra, depicting a straight line when graphed on a coordinate plane.
It symbolically represents the relationship between two variables, usually denoted as "x" and "y," via a first-degree polynomial equation.
The general form of a linear equation is written as:\[ Ax + By = C \]where \(A\), \(B\), and \(C\) are constants. However, the most commonly used format is the slope-intercept form, which is more intuitive when aiming to understand the geometry of the line on a graph.
  • Linear equations are characterized by their straight line shape.
  • The degree of the polynomials involved is always one, hence "linear."
  • Such equations can model real-world scenarios like speed over time, cost versus production, etc.
Every linear equation can be transformed into slope-intercept form where the role of each term becomes visibly clear.
Slope
The slope is a measure of the steepness or inclination of a line.
It quantifies how much the line rises or falls for each unit it moves horizontally along the x-axis.
In the slope-intercept form of a linear equation, represented as \(y = mx + b\), "m" denotes the slope.
  • Positive slope: Indicates an upward trajectory, moving from left to right.
  • Negative slope: Represents a downward path as you observe the line from left to right.
  • Zero slope: Indicates a perfectly flat, horizontal line.
  • Undefined slope: This occurs in vertical lines where the x-value remains constant, making division by zero impossible.
Calculating the slope from two points \((x_1, y_1)\) and \((x_2, y_2)\) involves the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Understanding slope is crucial for predicting how one variable changes in relation to another.
Y-Intercept
The y-intercept of a line is the point where the line meets the y-axis.
It provides an initial value or starting point when the x-value is zero.
In the slope-intercept form \(y = mx + b\), the "b" signifies the y-intercept.
  • Importance: The y-intercept acts as a constant in linear relationships, offering a benchmark from which changes occur.
  • Visual identification: Located directly on the y-axis, you can determine this value by setting \(x = 0\) in the equation.
In our example equation \(y = 4x - 3\), the y-intercept is \(-3\). This means the line intersects the y-axis at the point (0, -3).
Recognizing the y-intercept helps in sketching graphs effectively and in understanding the basic position of a line in a coordinate plane.
It often serves as an anchor point in storytelling data and establishing baseline metrics.

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Most popular questions from this chapter

Find the equation of the line that passes through the given point and also satisfies the additional piece of information. Express your answer in slope- intercept form, if possible. (-2,-7)\(;\) parallel to the line \(\frac{1}{2} x-\frac{1}{3} y=5\)

Solve each formula for the specified variable. $$C=2 \pi r \text { for } r$$

Find an equation of a line that passes through the point \((-A, B-1)\) and is perpendicular to the line \(A x+B y=C\) Assume that \(A\) and \(B\) are both nonzero.

Determine whether the lines are parallel, perpendicular, or neither, and then graph both lines in the same viewing screen using a graphing utility to confirm your answer. $$\begin{aligned} &y_{1}=17 x+22\\\ &y_{2}=-\frac{1}{17} x-13 \end{aligned}$$

Find the equation of the line that passes through the given point and also satisfies the additional piece of information. Express your answer in slope- intercept form, if possible. \((3,5),\) parallel to the \(y\) -axis
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