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Chapter 1: Problem 12

The value of the variable a solution to the equation? $$ x+3=x^{2}-9, x=4 $$

Short Answer

Expert verified
Answer: The value of 'a' is 1.

Step by step solution

01

Substitute the given x value into the equation

Begin by substituting the given x value, 4, into the equation which will allow us to simplify and solve for the variable 'a'. $$ 4 + 3 = a(4^{2} - 9) $$
02

Simplify the equation

Now, simplify the equation by solving the math operations within the parentheses and then proceed to isolate the variable 'a'. $$ 7 = a(16 - 9) $$
03

Solve for the variable 'a'

Continue simplifying and isolate the variable 'a' in the equation. $$ 7 = a(7) $$ Now, divide both sides of the equation by 7 to solve for 'a'. $$ a = \frac{7}{7} $$
04

Final solution

After simplifying the equation, we have found the value of the variable 'a'. $$ a = 1 $$ The value of the variable 'a' for the solution to the given equation is 1.

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Key Concepts

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

Solving Equations
To solve an equation means finding a value for the unknown that makes the equation true. In algebra, an equation is like a balance scale. Each side of the equation must have the same value. The equation given in this exercise is:
  • \( x + 3 = x^2 - 9 \)
Finding the solution involves several steps:
  • First, perform the operations that will isolate the unknown variable (like \( a \) in our context).
  • Simplify both sides of the equation to make equivalent comparisons.
Remember that solving equations often involves additional steps like combining like terms, distributing multiplication, or performing inverse operations. These steps can help you "isolate" the variable on one side.
Variable Substitution
Variable substitution is a technique where we replace a variable with a given number or expression. This method is often employed to simplify equations or evaluate expressions at a specific point.In the exercise, the value of \( x \) is given as 4. By substituting this value into the equation, we simplify and solve for \( a \). The substitution step changes the original equation:
  • Replace \( x \) with 4:
  • \( 4 + 3 = a(4^2 - 9) \)
This results in a simpler equation where we only need to focus on one variable. Substitution helps by turning a complex expression into something more manageable. It's vital to substitute accurately, as mistakes in this step can lead to incorrect solutions.
Simplifying Expressions
Simplifying expressions is a crucial skill in algebra that involves applying mathematical operations to make an expression easier to work with. This process was used in the solution to the original exercise as follows:The equation \( 7 = a(16 - 9) \) can be simplified further:
  • Calculate the expression in the parentheses: \( 16 - 9 = 7 \).
  • This simplifies the equation to: \( 7 = a \times 7 \).
The simplification makes it clear that the next step to find \( a \) is to divide both sides by 7.
  • \( a = \frac{7}{7} = 1 \)
Simplification often involves more than one type of operation, like factoring, expanding, or canceling common factors. It is essential to make a complex problem more approachable and to find solutions effectively.

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Most popular questions from this chapter

In Exercises \(24-27,\) write an expression for the sales tax on car. $$ \text { Tax rate is } r, \text { price is } \$ 20,000 \text { . } $$

Evaluate and simplify the expressions in Problems 44-53. $$ f(4) \text { given } f(x)=\frac{4 \sqrt{x-7}}{5-\sqrt{4 x-7}} $$

Oil well number 1 produces \(r_{1}\) barrels per day, and oil well number 2 produces \(r_{2}\) barrels per day. Each expression in Problems \(35-40\) describes the production at another well. What does the expression tell you about the well? $$ r_{1}-80 $$

The perimeter of a rectangle of length \(l\) and width \(w\) is \(2 l+2 w .\) Write a brief explanation of where the constants 2 in this expression come from.

In Exercises \(32-37,\) is \(y\) directly proportional to \(x ?\) If so, give the constant of proportionality. $$ x=5 y $$
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