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Chapter 0: Problem 15

Solve the equation and check your solution. $$ 8 x-5=3 x+20 $$

Short Answer

Expert verified
The solution to the equation \(8x - 5 = 3x + 20\) is \(x = 5\).

Step by step solution

01

Simplify the Equation

You can start by simplifying the equation on both sides. This involves eliminating like terms to leave the unknown on one side and the known on the other side. You can do this by subtracting \(3x\) from both sides and adding 5 to both sides to give: \(8x - 3x = 20 + 5\). This can be simplified further to give \(5x = 25\).
02

Solve for \(x\)

After simplifying the equation, you can solve for \(x\) by dividing both sides of the equation by 5. This gives: \(x = 25/5 = 5\).
03

Check the Solution

The final step is to validate your answer by substituting \(x = 5\)` into the original equation. Substituting \(x\) back into the equation gives: \(8*5 - 5 = 3*5 + 20\), which simplifies to \(35 = 35\). As both sides of the equation are equal, the answer is correct.

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Key Concepts

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

Solving Equations
Solving equations can seem intimidating at first, but breaking it down into simple steps can make it much easier. An equation is like a balance scale. Both sides need to be equal for the equation to 'balance' correctly. In our scenario, we want to find the value of \(x\) that makes both sides of the equation \(8x - 5 = 3x + 20\) equal.

To solve an equation effectively, the first move is usually to simplify it by getting rid of any unnecessary terms on each side. This means adjusting each side so only like terms are positioned on either side of the equal sign. By doing this, you're setting up the stage for algebraic manipulation to isolate the variable. Let's look at this further in our next sections.
Algebraic Manipulation
Algebraic manipulation involves applying basic arithmetic operations to both sides of an equation to rearrange terms. Let's examine our equation:
  • Start with the given equation: \(8x - 5 = 3x + 20\).
  • Our first goal is to get all \(x\) terms on one side. We achieve this by subtracting \(3x\) from both sides of the equation. Now it looks like this: \(8x - 3x - 5 = 20\).
  • Now simplify further: \(5x - 5 = 20\).
  • Next, we'll get rid of the \,\(-5\) term on the left by adding \(5\) to both sides, leading to \(5x = 25\).
With the equation simplified to this form, we are just a step away from finding the solution. Algebraic manipulation helps bring us closer to isolating the desired variable by using arithmetic operations judiciously.
Substitution Method
The substitution method involves checking our solution to confirm that we solved the equation correctly. While our step-by-step solution gave us \(x = 5\), it's vital to ensure this value satisfies the original equation.

To do this, substitute the found value back into the original equation:
  • Replace \(x\) in \(8x - 5\) with \(5\), getting \(8(5) - 5 = 40 - 5 = 35\).
  • Do the same for \(3x + 20\), leading to \(3(5) + 20 = 15 + 20 = 35\).
Both calculations yield \(35\), demonstrating that our solution is correct and both sides of the equation are balanced. This validates the process of solving an equation by confirming the integrity of our solution.

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Most popular questions from this chapter

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ g^{-1} \cdot f^{-1} $$

In Exercises 69-74, use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ f^{-1}\right)(-3) $$

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ f(t)=t^{2}+2 t-3 $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=x^{3}-3 x^{2}-x &\quad x_{1}=1, x_{2}=3 \end{array} $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=3 x+8 \quad & x_{1}=0, x_{2}=3 \end{array} $$
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