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Chapter 7: Problem 2

Graph the equations and inequalities. $$ 4 x-3=-7 $$

Short Answer

Expert verified
Answer: The graph of the equation 4x - 3 = -7 is a vertical line that passes through the point (-1,0).

Step by step solution

01

Solve the equation for x

To find the value of x, we will first isolate the x-term by adding 3 to both sides of the equation: $$ 4x - 3 + 3 = -7 + 3 \\ 4x = -4 $$ Now, we will divide both sides of the equation by 4 to solve for x: $$ \frac{4x}{4} = \frac{-4}{4} \\ x = -1 $$
02

Graph the equation on a coordinate plane

Since we are dealing with a linear equation, the graph will be a straight line. However, because there is only one variable (x), the line is vertical and passes through the point (-1,0). To graph the equation, draw a vertical line that crosses the x-axis at -1. The line should extend infinitely up and down, as it represents all possible values of y for x = -1.

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Key Concepts

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

Linear Equations
Linear equations are mathematical expressions that represent straight lines when graphed on a coordinate plane. They are generally written in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. This type of equation involves a single variable that is not raised to any power higher than one. Given their simplicity, linear equations are an essential foundation for understanding more complex algebraic concepts.
  • When solving, the aim is to find the value of the variable that makes the equation true.
  • In the equation \( 4x - 3 = -7 \), we will manipulate the terms to isolate \( x \).
Linear equations have a predictable slope and intercept, which makes them easy to graph and interpret visually. Grasping these fundamentals will help as you move on to solving and graphing them.
Solving Equations
Solving equations is a process where we find the value of the variable that satisfies the equation. In the linear equation \( 4x - 3 = -7 \), solving involves isolating the variable \( x \) to determine its numeric value.
Start by simplifying both sides. Here, we add 3 to both sides to remove the constant term attached to \( x \):\[4x - 3 + 3 = -7 + 3 \4x = -4\]Then, divide both sides by 4:\[\frac{4x}{4} = \frac{-4}{4} \x = -1\]The solution \( x = -1 \) tells us the x-coordinate where the line intersects the x-axis. Always check your work by substituting \( x = -1 \) back into the original equation to confirm.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can graph equations and visualize their solutions. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis.
Each point on the plane is identified by an ordered pair \((x, y)\), which describes its horizontal and vertical positions.
  • The origin \((0, 0)\) is where these axes intersect.
  • Quadrants divide the plane into four regions based on the signs of \( x \) and \( y \).
For this exercise, the variable \( x = -1 \) represents a vertical line on the coordinate plane. Visualizing equations in this way helps us better understand relationships between variables and the nature of their solutions.
Graphing Techniques
Graphing techniques are strategies used to represent equations visually. For linear equations like those involving only \( x \) or \( y \), graphing can be straightforward yet powerful.
  • Begin by identifying key points that satisfy the equation. In this case, identify \( x = -1 \).
  • Draw a vertical line through the x-coordinate \( -1 \) extending infinitely up and down.
This represents all x-values where the equation holds true. Graphs can show solutions more visually and indicate whether equations have unique, infinite, or no solutions. Pay attention to steepness, direction, and intersection points to garner further insights from your graph.

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Most popular questions from this chapter

Determine the slope and \(y\) -intercept of the lines. $$ y=x $$

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

Graph the equations. $$ x=4 $$

Determine the slope and \(y\) -intercept of the lines. $$ 7 y+3 x=10 $$

The slope of a straight line is a _______ of the steepness of the line.
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