Problem 48 - Algebra and Trigonometry

Question

write each English sentence as an equation in two variables. Then graph the equation. The \(y\) -value is the difference between four and twice the \(x\) -value.

Step-by-Step Solution

Answer

The equation corresponding to the English sentence is \(y = 4 - 2x\). The graph of this equation is a straight line that passes through points (0,4) and (2,0).

Step 1: Translate English sentence into a mathematical equation

From 'The \(y\) -value is the difference between four and twice the \(x\) -value' we derive the equation: \(y = 4 - 2x\). In this equation, \(y\) is expressed as being equal to four minus twice the value of \(x\)

Step 2: Graph the Equation

In order to graph the equation \(y = 4 - 2x\), remember that it's a linear equation and its graph will therefore be a straight line. Start by finding the y-intercept by setting \(x = 0\), which results in \(y = 4\). This gives the point (0,4). Now find the x-intercept by setting \(y = 0\), which results in \(x = 2\). This gives the point (2,0). Plot these two points on a graph and draw a line through them.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a foundational concept in algebra that involves plotting lines on a coordinate plane. A linear equation can be recognized by its simplest form, which is often written as \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) is the y-intercept. When graphing, you plot points on a graph and connect them with a straight line. For example, in the equation \( y = 4 - 2x \), you need to identify suitable points that lie on the line.

To graph any linear equation properly, follow these steps:

• Find the y-intercept by setting \( x = 0 \).
• Find the x-intercept by setting \( y = 0 \).
• Plot these intercepts on the coordinate plane.
• Draw a straight line through these plotted points extending in both directions.

Remember that because this is a linear equation, the graph is always a straight line. This makes it easier to predict the slope and orientation of the line
and confirm your plot is correct.

Two-Variable Equations

Two-variable equations are expressions that involve two different variables, typically \( x \) and \( y \). These equations form straight lines when graphed on a coordinate grid, thus reflecting linear relationships. In our example where the equation is \( y = 4 - 2x \), \( x \) and \( y \) are the variables.

The value of \( y \) is dependent on the value of \( x \), which dictates the position of points on the line. In the case of the equation stemming from the exercise, every change in \( x \) affects \( y \) due to the negative 2x term, which ladders the line downward. Understanding two-variable equations helps in solving problems that require relationships between distinct values or measurements.

The key characteristics of these equations include:

• They have two independent variables, often \( x \) and \( y \).
• Their solutions form a straight line on a graph.
• The slope \( m \) indicates the rate of change of \( y \) with respect to \( x \).
• The intercept \( b \) gives a starting point on the y-axis when \( x = 0 \).

This framework allows you to discern not just single points but the entire trajectory of two-variable equations in a coordinate plane.

Y-Intercept and X-Intercept

The y-intercept and x-intercept are crucial characteristics of lines on a graph pertaining to linear equations. They help in understanding where a line crosses the axes, enabling easier plotting and comprehension of a line's behavior.

The y-intercept is the point where the line crosses the y-axis. It represents the value of \( y \) when \( x = 0 \). For the equation \( y = 4 - 2x \), the y-intercept is 4, which is denoted by the point (0, 4) on the graph.

On the other hand, the x-intercept is where the line crosses the x-axis. This occurs when \( y = 0 \). In our example, solving for \( x \) when \( y \) is zero gives \( x = 2 \), marking the x-intercept at (2, 0).

Here are the important pointers:

• The y-intercept tells how high the line starts on the y-axis when it intercepts.
• The x-intercept informs where the line crosses the x-axis.
• Intercepts aid in deciding the position and direction of the line graphically.
• Calculating intercepts is a straightforward process: set one variable to 0 and solve the equation for the other.

Utilizing intercepts efficiently allows you to sketch graphs rapidly, verifying the correctness of plotted lines by aligning them with these major points of intersection.