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Chapter 2: Problem 4

For exercises 1-72, (a) solve. (b) check. $$ 3 x-15=36 $$

Short Answer

Expert verified
x = 17

Step by step solution

01

Isolate the variable term

Add 15 to both sides of the equation to isolate the term with the variable.
02

Simplify the equation

Simplify the left and right side of the equation: \[ 3x - 15 + 15 = 36 + 15 \] simplifying, we get \[ 3x = 51 \]
03

Solve for x

Divide both sides by 3 to solve for \( x \): \[ x = \frac{51}{3} \] Simplifying, we get \[ x = 17 \]
04

Check the solution

Substitute \( x = 17 \) back into the original equation and verify: \[ 3(17) - 15 = 36 \] This simplifies to \[ 51 - 15 = 36 \] which further simplifies to \[ 36 = 36 \], confirming our solution is correct.

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Key Concepts

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

Isolating the Variable
Isolating the variable is the first step in solving linear equations. The goal is to move the variable term to one side of the equation, keeping it by itself. This means that all other numbers (constants) need to be moved to the opposite side of the equation using basic arithmetic operations. In the given exercise, our objective is to isolate the term with the variable \(x\).
  • Start with the original equation: \[3x - 15 = 36 \]
  • We need to eliminate the \-15\ next to \(3x\). To do this, add 15 to both sides of the equation, which gives us: \[3x - 15 + 15 = 36 + 15 \]
The idea is to perform the same operation on both sides to keep the equation balanced. This step ensures that the variable is on one side by itself, making it easier to solve.
Simplifying Equations
After isolating the variable, the next step is to simplify the equation. Simplifying makes the equation easier to solve and helps in getting clear results. In our case, after adding 15 to both sides, we need to perform the addition:
  • Simplify \[3x - 15 + 15 = 36 + 15 \] to get \[3x = 51 \].
This means we have successfully moved the constant to the other side and simplified both sides. Now, we can solve for \(x\) by further isolating it through division.
Checking Solutions
Checking the solution is a crucial final step in solving linear equations. It ensures that the value obtained for the variable makes the original equation true. We need to substitute the value of \(x\) back into the original equation to verify:
  • Our solution is \(x = 17\).
  • Substitute \(x = 17\) back into the original equation: \[3(17) - 15 = 36 \]
  • Simplify inside the parentheses: \[51 - 15 = 36 \]
Since both sides of the equation equal 36, our value for \(x\) is correct. This verification step checks if the operations were performed correctly and confirms that the solution meets the equation's requirements.

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Most popular questions from this chapter

Solve \(R=\frac{E+C}{L}\) for \(E\)

Solve \(V=\frac{L W T}{144}\) for \(W\).

For exercises 9-36, (a) solve. (b) check the direction of the inequality sign. $$ -p+12>3 $$

a. Solve \(D=\frac{M}{V}\) for \(V\). b. The density \(D\) of gasoline is \(\frac{0.737 \mathrm{~g}}{1 \mathrm{~mL}}\), and the mass \(M\) is \(50 \mathrm{~g}\). Find the volume, \(V\). Round to the nearest tenth.

For exercises 29-36, solve the equation for \(x\). Solve \(L=C-H-\frac{1}{5} T\) for \(C\).
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