Problem 47 - College Algebra Essentials

Question

Write each English sentence as an equation in two variables Then graph the equation. The \(y\) -value is four more than twice the \(x\) -value.

Step-by-Step Solution

Answer

The equivalent equation for the sentence is \(y = 2x + 4\) and the graph is a straight line with slope 2 and y-intercept 4.

Step 1: Translate Sentence into Math Equation

First, is to translate the English sentence into mathematics language. 'The \(y\) -value is four more than twice the \(x\) -value' can be translated into \(y = 2x + 4\) . 'Four more than twice the \(x\) -value' means that we first multiply \(x\) by 2, and then add 4 to the result.

Step 2: Graph the Equation

With the obtained equation \(y = 2x + 4\) , it can be recognized as a linear equation in slope-intercept form. The slope of the line is 2 and the y-intercept is 4. This means that the line crosses the y-axis at the point (0,4), and for every unit increase in \(x\) , \(y\) increases by 2 units. To graph it, start by plotting the y-intercept at point (0,4). Then from that point, rise 2 units and run 1 unit to the right, this gives the second point (1,6). Connect these two points with a straight line extending in both directions.

Key Concepts

Linear Equations in Two Variables

Understanding linear equations in two variables is a fundamental concept in algebra that sets the stage for more complex mathematical explorations. A linear equation can always be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are the variables. Such an equation represents a straight line when graphed on a two-dimensional plane. The reason this type of equation is called 'linear' is due to it forming a line when graphed.

To sketch the graph of such an equation, you'll need to identify at least two points on the line that satisfy the equation. Once you plot those points on Cartesian coordinates (also known as an 'xy' graph), you simply draw a straight line through them, and that represents the solution set to the equation - every point on that line is a solution to the equation.

For students beginning to learn this concept, it can be helpful to work with specific examples and plot points to see the linear relationship between the two variables. As you become more comfortable with plotting points and drawing lines, you will find that graphing linear equations can be both intuitively satisfying and an essential skill for solving real-world problems.

Slope-Intercept Form

The slope-intercept form of a linear equation is a special way of writing the equation to make graphing easier. It is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope \(m\) indicates how steep the line is, and the y-intercept \(b\) tells you where the line crosses the y-axis.

Significance of Slope and Intercept

Understanding the slope is crucial because it describes the direction and steepness of the line. A positive slope means the line is ascending, while a negative slope means it's descending. The y-intercept is equally important because it provides a starting point for drawing your line on the graph.

To graph an equation in slope-intercept form, begin by plotting the y-intercept on the graph. This would be the point where \(x=0\). After that, use the slope to determine the line's direction and steepness. If the slope is 2, as in our example (\(y = 2x + 4\)), you would move up two units for every one unit you move to the right. If our slope were a fraction, such as \(1/2\), then for every one unit we move to the right, we would only move up half a unit. By using the slope and the y-intercept together, graphing the line becomes a manageable task that, once mastered, can help solve more extensive and more complex math problems.

Translating English to Math

The process of translating English to math is an essential skill that enables students to transform real-world scenarios into mathematical equations. This translation often involves recognizing keywords and phrases that correspond to mathematical operations and relationships.

Keywords and Phrases

In our exercise, the phrase 'four more than twice the \(x\)-value' indicates an operation. 'Twice' suggests multiplication by two, and 'four more than' indicates addition. Thus, the English sentence is converted into the equation \(y = 2x + 4\). Such translations are crucial for setting up and solving problems across various fields of mathematics and science.

Practice with translation helps build a bridge between the language of mathematics and everyday communication, making it easier to approach and solve problems. As students work on these translation skills, they will find it increasingly natural to convert phrases into equations, making them more confident and proficient in their mathematical abilities. It's a skill that proves beneficial not only in mathematics but in any domain where quantitative reasoning is required.