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Chapter 2: Problem 47

Write each equation in \(y=b\) or \(x=a\) form by solving for \(y\) or \(x\). Then graph it. \(-2 x+3=11\)

Short Answer

Expert verified
The equation is \(x = -4\) and it's a vertical line at \(x = -4\).

Step by step solution

01

Isolate the variable term

Start with the given equation: \(-2x + 3 = 11\)Your goal is to solve for \(x\). First, we need to isolate the term that contains \(x\). To do this, subtract 3 from both sides of the equation. \(-2x + 3 - 3 = 11 - 3\)This simplifies to:\(-2x = 8\)
02

Solve for the variable

With \(-2x = 8\), we want to solve for \(x\). Divide both sides by -2 to isolate \(x\):\(x = \frac{8}{-2}\)This simplifies to:\(x = -4\)
03

Express the equation in desired form

The equation is now solved for \(x\) and can be expressed as:\(x = -4\)This is already in the \(x = a\) form, where \(a = -4\).
04

Graph the equation

The equation \(x = -4\) represents a vertical line that passes through \(x = -4\) on the x-axis. On a graph, draw a straight, vertical line through the point \((-4, y)\) for all values of \(y\). This line does not intersect the y-axis and is parallel to the y-axis.

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Key Concepts

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

Isolating Variables
In algebra, isolating variables is an essential process and a common step in solving equations. The primary objective is to manipulate the equation until the variable of interest stands alone on one side of the equation.
  • Identify the variable you need to isolate, which is the term you need to separate from the rest of the equation.
  • Perform operations to both sides of the equation to remove any constants (numbers) or coefficients (factors multiplying the variable) from the variable side.
  • Ensure to maintain equality by performing the same operation on both sides of the equation, using addition, subtraction, division, or multiplication as needed.
In our original problem, \(-2x + 3 = 11\) involved isolating \(x\). We began by subtracting 3 from both sides of the equation to rid \(x\) of any constants, simplifying to \(-2x = 8\). We then divided both sides by \(-2\) to isolate \(x\), resulting in \(x = -4\). Each of these steps allowed us to systematically remove any excess components clinging to the variable, ensuring \(x\) stands independently.
Graphing Vertical Lines
Graphing equations is a pivotal skill in visualizing algebraic solutions, with vertical lines presenting a unique case on the Cartesian plane.
Vertical lines have a distinct method of graphing since they denote all points that share the same x-coordinate.
  • Vertical lines are constant along the x-axis, meaning they contain points only of the form \((a, y)\).
  • These lines do not intersect the y-axis, remaining parallel to it instead.
  • To graph a vertical line like \(x = -4\), draw a straight line passing through \(-4\) on the x-axis. This line should stretch upward and downward indefinitely while all the y-values of points on this line can vary.
This property shows that vertical lines illustrate scenarios where the x-value remains constant and unchanging despite differences in y-values.
Equation Forms
Equations can be expressed in different forms, each serving a specific purpose and providing distinct insights. Two notable forms are **\(y = b\)** and **\(x = a\)**.
  • \(y = b\)** denotes a horizontal line where y is constant and represents all points with coordinates of the form \((x, b)\).
  • \(x = a\)** describes vertical lines, as in our original solution of \(x = -4\), where all points share the same x-coordinate \(a\).
While solving the earlier equation resulted in \(x = -4\), adopting the \(x = a\) form helps identify the nature of the line鈥攙ertical in this instance鈥攁nd outlines its graphical representation efficiently. Thus, equation forms simplify the understanding of graph behavior and the specific nature of the relationship between variables.

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Most popular questions from this chapter

Explain how to graph a function by plotting points.

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