Problem 39 - Thinking Mathematically

Question

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(7 x+2 y=14\)

Step-by-Step Solution

Answer

The slope-intercept form of the given equation is \(y = -\frac{7}{2}x + 7\). The slope of the line is -7/2 and the y-intercept is 7.

Step 1: Put the Equation in Slope-Intercept Form

The given equation is \(7x + 2y = 14\). The slope-intercept form of an equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To transform the given equation into this form, you need to isolate \(y\). Start by subtracting \(7x\) from both sides to get \(2y = -7x + 14\). Then divide every term by 2 to isolate \(y\), resulting in \(y = -\frac{7}{2}x + 7\).

Step 2: Identify the Slope and Y-Intercept

From the slope-intercept equation, \(y = -\frac{7}{2}x + 7\), it is seen that the slope \(m\) is \(-\frac{7}{2}\) and the y-intercept \(b\) is 7. Therefore the slope is -7/2 and the y-intercept is 7.

Step 3: Graph the Line Using the Slope and Y-Intercept

Start at the y-intercept (0,7) on the graph. The slope \(-\frac{7}{2}\) means move down 7 units (negative sign indicates downward direction) and move to the right 2 units for every step on the graph. Plot these points and draw the line which goes through them.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. The purpose of algebra is to find the unknown values that make an equation true, which is known as solving the equation. In the context of linear equations, algebra is used to manipulate an equation to get it into a form that is easier to understand and work with, such as the slope-intercept form.

Using algebraic techniques to solve for variable 'y' in the equation '7x + 2y = 14,' demonstrates a practical application of these principles. The solution involves rearranging terms and simplifying the equation step by step, which is a fundamental skill in algebra. This skill enhances the students' ability to approach and solve various types of equations they will encounter in their studies.

Graphing Linear Equations

Graphing linear equations involves plotting a straight line onto a coordinate plane that represents all the solutions to the equation. This visual representation is crucial because it helps students understand the relationship between variables. When a linear equation is in slope-intercept form, such as the equation \(y = -\frac{7}{2}x + 7\), it is particularly easy to graph.

The line's steepness and direction are indicated by the slope, while its starting position on the y-axis is given by the y-intercept. Plotting the y-intercept is the starting point, and from there, you use the slope to find other points on the line. In educational settings, graphing is a pivotal skill that helps students visualize the concepts they learn in algebra.

Slope and Y-Intercept

The slope and y-intercept are two key characteristics of a linear equation when expressed in its slope-intercept form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.

The slope, calculated as 'rise over run,' determines the angle and the direction of the line on the graph. A positive slope slants upward, while a negative slope, like \( -\frac{7}{2}\), slants downward. The y-intercept represents the point where the line crosses the y-axis. In the provided exercise, the y-intercept is 7, which means the line crosses the y-axis at the point \(0, 7\). Knowing how to identify and use these elements enables students to accurately draw the graph of any linear equation quickly and efficiently.