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Chapter 6: Problem 22

Solve and check each equation. \(10(3 x+2)=70\)

Short Answer

Expert verified
The solution of the equation is \(x = 5/3\).

Step by step solution

01

Distribute

First, distribute the 10 to both terms inside the parentheses: \(10 * 3x + 10 * 2 = 70\), which simplifies to \(30x + 20 = 70\).
02

Simplify equation

Next, isolate 'x' on one side of the equation by subtracting 20 from both sides: \(30x = 70 - 20\). This simplifies to \(30x = 50\).
03

Solve for 'x'

Divide both sides of the equation by 30 to solve for 'x': \(x = 50/30\). And this simplifies to \(x = 5/3\).
04

Check the Solution

To check this solution, substitute \(5/3\) back into the original equation \(10(3x + 2)=70\). This gives us \(10(3*(5/3) + 2) = 10(5 + 2) = 70\), which does equals to 70, the right-hand side of the original equation. Therefore, \(x = 5/3\) is indeed a correct solution.

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that helps simplify expressions. It allows us to multiply a single term by each term within a set of parentheses. In our exercise, we apply the distributive property to the equation \(10(3x + 2) = 70\).

Here's how it works: multiply 10 by each term inside the parentheses:
  • 10 times \(3x\) gives \(30x\)
  • 10 times 2 gives 20
Thus, the expression \(10(3x + 2)\) simplifies to \(30x + 20\). The distributive step transforms a compact expression into one that's easier to work with, setting the stage for simplifying the rest of the equation.
Simplifying Equations
Simplifying equations involves reducing the expression to its simplest form. After using the distributive property, we have \(30x + 20 = 70\). The goal is to make the equation cleaner, focusing on isolating the variables.

To simplify:
  • Subtract 20 from both sides to eliminate the constant term from the left side: \(30x = 70 - 20\).
  • This results in \(30x = 50\), a much simpler equation where \(x\) is now closer to being isolated.
By following these steps, we clear away unnecessary terms, speeding up the process of solving for \(x\).
Isolating Variables
Isolating the variable is the key step in solving an equation: it involves getting the variable alone on one side. In our simplified equation, \(30x = 50\), we need to isolate \(x\).

To do this:
  • Divide both sides of the equation by 30, the coefficient of \(x\).
  • This gives \(x = 50/30\), simplifying further to \(x = 5/3\).
By isolating \(x\), we determine its value, finishing the equation-solving process. This step is crucial as it delivers the solution we seek.
Checking Solutions
Checking the solution ensures that the value calculated for the variable actually satisfies the original equation. We do this by substituting \(x = 5/3\) back into the original equation \(10(3x + 2)=70\).

Here's how you check:
  • Replace \(x\) with \(5/3\) to get \(10(3(5/3) + 2)\).
  • Simplify this step by step: \(3(5/3) = 5\), then \(5 + 2 = 7\).
  • Thus, the expression becomes \(10 \times 7 = 70\).
Since both sides of the equation equal 70, the solution \(x = 5/3\) is correct. This verification step is essential, as it confirms the solution鈥檚 validity.

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Most popular questions from this chapter

Solve the quadratic equations in Exercises 37-52 by factoring. \(x^{2}-12 x=-36\)

In Exercises 33-36, solve each equation using the zero-product principle. \((x-8)(x+3)=0\)

The fastest way for me to solve \(x^{2}-x-2=0\) is to use the quadratic formula.

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(6 x^{2}-17 x+12\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}+8 x+15=0\)
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