Problem 83 - Introductory Algebra for College Students

Question

write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable exceeds twice the \(x\) -variable by 5

Step-by-Step Solution

Answer

The linear equation is \(y = 2x + 5\). The graph of the equation is a straight line crossing through the points (0,5) and (1,7).

Step 1: Translate sentence into equation

Based on the sentence, we can express it mathematically as 'y = 2x + 5'. This sentence translates to 'y is equal to twice x plus 5'.

Step 2: Graph the equation

To graph the linear equation \(y = 2x + 5\), we need at least two points. - Begin with x = 0, we will get y value, y = 2*(0) + 5 = 5. So, we have the point (0,5). - Next, choose x = 1. Substituting x = 1 into the equation, we find y = 7. So, this gives us the second point (1,7). Use the 2 points (0,5) and (1,7) to draw a straight line, which graphically represents this equation.

Key Concepts

Linear Equations in Two Variables

Understanding linear equations with two variables is crucial in algebra and in various real-world applications. A linear equation is an algebraic equation where any term is either a constant or the product of a constant and the first power of a single variable.

Linear equations in two variables typically take the form of \( y = mx + b \), where \( m \) and \( b \) are constants, and \( x \) and \( y \) are variables. The constant \( m \) is known as the slope of the line which describes how steep the line is, while \( b \) represents the y-intercept, the point where the line crosses the y-axis.

In real life, these equations can represent relationships where one quantity depends on another, such as distance over time or supply related to demand. It's important to get comfortable with identifying and graphing these equations to visualize the relationships between variables.

Translating Sentences into Algebraic Expressions

When faced with a problem involving verbal descriptions, translating sentences into algebraic expressions is an essential skill. It is about converting words into symbols that represent mathematical operations and relationships.

For example, in the given exercise, the phrase 'the \(y\)-variable exceeds twice the \(x\)-variable by 5' is translated to the algebraic expression \(y = 2x + 5\). Here, 'exceeds' suggests addition, 'twice' indicates multiplication by two, and 'by 5' denotes that five is added to the product of two and the \(x\)-variable.

Tips for Translation:

• Identify the operations described by the words: addition ('sum' or 'more than'), subtraction ('difference' or 'less than'), multiplication ('twice' or 'product'), and division ('ratio' or 'quotient').
• Look for clue words that indicate mathematical operations.
• Convert phrases into variable expressions methodically, maintaining the order of operations.

The ability to perform these translations correctly leads to accurate algebraic expressions, setting the stage for further mathematical operations.

Plotting Points on a Cartesian Plane

A Cartesian plane, named after René Descartes, is used to plot points and visualize relationships between them. It consists of two perpendicular number lines: the horizontal axis known as the x-axis, and the vertical axis known as the y-axis. The intersection where these axes meet is called the origin, with coordinates (0,0).

When plotting points, each point is defined by a pair of numbers written as \((x, y)\), where \(x\) corresponds to the horizontal position and \(y\) the vertical. To plot a point, start at the origin, move along the x-axis by the first number and then parallel to the y-axis by the second number.

In the exercise solution, the points (0,5) and (1,7) are plotted on the Cartesian plane. To draw a line, simply connect these points with a straight edge.

Steps to Plot a Point:

• Locate the x-coordinate on the x-axis.
• From there, move vertically to the y-coordinate.
• Mark the spot where this vertical line from the x-coordinate intersects with the y-coordinate level.

Practicing this process helps create a visual representation of linear equations and other relationships, which can be hugely beneficial for interpreting data and understanding mathematical concepts.