Problem 27
Question
Find the \(y\) -intercept of the line. $$ y=4 x-2 $$
Step-by-Step Solution
Verified Answer
The y-intercept of the line is -2.
1Step 1: Identify the y-intercept from slope-intercept form
The equation \(y = 4x - 2\) is in slope-intercept form \(y = mx + b\), where \(b\) is the \(y\)-intercept.
2Step 2: State the answer
The \(y\)-intercept is \(b = -2\). The line crosses the \(y\)-axis at the point \((0, -2)\).
Key Concepts
Linear EquationsSlope-Intercept FormAlgebraic Expressions
Linear Equations
A linear equation is an algebraic expression that represents a straight line when graphed on the coordinate plane. These equations typically take the form of \( y = mx + b \), where \( m \) stands for the slope of the line, and \( b \) represents the y-intercept. The slope indicates the steepness or angle of the line, while the y-intercept indicates the point at which the line crosses the y-axis.
Understanding linear equations is crucial because they are used to describe relationships where one variable changes at a constant rate with respect to another. In practical terms, linear equations can represent trends in data, such as speed over time, or cost over quantity. The ability to solve these equations helps in various scientific, economic, and mathematical contexts. When analyzing a linear equation like \( y = 4x - 2 \) from the original exercise, recognize that it describes a line that increases by 4 units in the y direction for every 1 unit it moves in the x direction and crosses the y-axis at -2.
Understanding linear equations is crucial because they are used to describe relationships where one variable changes at a constant rate with respect to another. In practical terms, linear equations can represent trends in data, such as speed over time, or cost over quantity. The ability to solve these equations helps in various scientific, economic, and mathematical contexts. When analyzing a linear equation like \( y = 4x - 2 \) from the original exercise, recognize that it describes a line that increases by 4 units in the y direction for every 1 unit it moves in the x direction and crosses the y-axis at -2.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common ways to express these equations. It is written as \( y = mx + b \) and offers a straightforward portrayal of a line's characteristics, directly revealing the slope (\( m \) ) and the y-intercept (\( b \) ).
In this form, \( m \) captures how the line tilts or inclines and is calculated by the rise over run, that is, the change in y divided by the change in x between any two points on the line. The y-intercept \( b \) is simply the value of y when \( x \) is zero. This form is convenient for graphing lines and solving various problems involving linear relationships because you can easily determine key characteristics of the line at a glance.
For the equation \( y = 4x - 2 \) given in the exercise, \( 4 \) is the slope, and \( -2 \) is the y-intercept, indicating that for every unit increase in \( x \) , \( y \) increases by 4 units, and the line crosses the y-axis below the origin at \( y = -2 \).
In this form, \( m \) captures how the line tilts or inclines and is calculated by the rise over run, that is, the change in y divided by the change in x between any two points on the line. The y-intercept \( b \) is simply the value of y when \( x \) is zero. This form is convenient for graphing lines and solving various problems involving linear relationships because you can easily determine key characteristics of the line at a glance.
For the equation \( y = 4x - 2 \) given in the exercise, \( 4 \) is the slope, and \( -2 \) is the y-intercept, indicating that for every unit increase in \( x \) , \( y \) increases by 4 units, and the line crosses the y-axis below the origin at \( y = -2 \).
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. These expressions can vary in complexity from simple forms like \( 3x + 2 \) to more complex polynomial expressions. Unlike equations, algebraic expressions do not have an equality sign; they are not statements that claim two things are the same, but rather they simply represent a quantity.
Algebraic expressions are essential in forming equations that model real-world scenarios. The process of writing and simplifying these expressions is foundational in algebra. Knowing how to manipulate and evaluate them is key, whether you are solving for a variable or substituting numbers in to find specific values.
In our example, \( y = 4x - 2 \) could be considered an algebraic expression on the right side. The expression tells us how to compute the value of \( y \) for any given value of \( x \) and is critical for understanding the relationship between the variables in the context of the equation.
Algebraic expressions are essential in forming equations that model real-world scenarios. The process of writing and simplifying these expressions is foundational in algebra. Knowing how to manipulate and evaluate them is key, whether you are solving for a variable or substituting numbers in to find specific values.
In our example, \( y = 4x - 2 \) could be considered an algebraic expression on the right side. The expression tells us how to compute the value of \( y \) for any given value of \( x \) and is critical for understanding the relationship between the variables in the context of the equation.