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Chapter 7: Problem 6

Graph the linear equations and inequalities. $$ 4 x+7=19 $$

Short Answer

Expert verified
Answer: The slope (m) is 4, and the y-intercept (b) is 19.

Step by step solution

01

Solve for x

To isolate x, we'll begin by subtracting 7 from both sides of the equation: $$ 4x + 7 - 7 = 19 - 7 \\ 4x = 12 $$ Next, divide both sides by 4: $$ \frac{4x}{4} = \frac{12}{4} \\ x = 3 $$
02

Identify the slope and y-intercept

The given equation is $$4x + 7 = 19$$, which is in the standard form (Ax + By = C). To determine the slope (m) and y-intercept (b), we need to rewrite it in slope-intercept form (y = mx + b). Since x = 3, we have: $$ y = 4(3) + 7 \\ y = 12 + 7 \\ y = 19 $$ In this case, the slope (m) is 4, and the y-intercept (b) is 19.
03

Graph the line

Using the slope(m) and the y-intercept(b), plot the linear equation on the graph. First, plot the y-intercept at the point (0,19). Then, using the slope of 4, move up 4 units and to the right 1 unit to find another point (1,23). Draw a line through these two points representing the equation $$y = 4x + 7$$.
04

Shade the area representing the inequality

Since we only have a linear equation, there is no inequality to shade on the graph. If there was an inequality symbol, such as <, >, 鈮�, or 鈮�, we would shade the region above or below the line accordingly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Linear Equations
Graphing linear equations is a fundamental skill in mathematics. It involves plotting the solutions of an equation on a coordinate plane. In the case of linear equations, the graph is always a straight line. A linear equation typically follows the form \(Ax + By = C\), where the relationship between \(x\) and \(y\) represents a linear relationship.
The first step in graphing a linear equation is identifying the coordinates of points that satisfy the equation. These points, when plotted and connected, form the line that represents the equation.
  • Plot the y-intercept: This is where the line crosses the y-axis. It's a starting point for drawing the line.
  • Use the slope to find another point: The slope, often denoted "m," shows how steep the line is. It can tell you how to extend the line from the y-intercept.
By plotting these points and drawing a line through them, you've graphed the linear equation.
Slope-Intercept Form
The slope-intercept form is a way of writing linear equations to make it easy to graph them. This form is written as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.
Understanding these components:
  • Slope (m): It tells us how much \(y\) changes with a change in \(x\). A greater absolute value of the slope indicates a steeper line.
  • Y-intercept (b): This value is where the line crosses the y-axis. It gives you the starting point to graph the line.
To convert an equation to slope-intercept form, solve for \(y\) to isolate it on one side of the equation. This provides a clear view of how \(x\) and \(y\) relate to each other, facilitating direct plotting.
Solving Equations
Solving linear equations involves finding the value of the variable that makes the equation true. It is a crucial skill in algebra that uses logical steps to isolate the variable on one side of the equation.
Here's how you do it for the equation \(4x + 7 = 19\):
  • Subtract 7 from both sides: \(4x + 7 - 7 = 19 - 7\) simplifies to \(4x = 12\).
  • Divide by 4: \(\frac{4x}{4} = \frac{12}{4}\) simplifies to \(x = 3\).
The solution is the value of \(x\) that makes the equation true. Once \(x\) is known, it can be substituted back into the original equation to verify the solution. Solving equations often requires a series of arithmetic operations like addition, subtraction, multiplication, and division to isolate the variable.

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Most popular questions from this chapter

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form. slope \(=7, \quad\) passesthrough (0,0) .

Draw a coordinate system and plot the following ordered pairs. $$ (3,1),(4,-2),(-1,-3),(0,3),(3,0),\left(5,-\frac{2}{3}\right) $$

Write the equation of the line using the given information. Write the equation in slope-intercept form. Slope \(=3, \quad y\) -intercept \(=-6\)

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form. passes through the points (5,2) and (2,1) .

Graph the equations. $$ x=0 $$
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