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Chapter 1: Problem 2

In Exercises \(1-16,\) solve and check each linear equation. $$ 6 x-3=63 $$

Short Answer

Expert verified
The solution to the linear equation \(6x - 3 = 63\) is \(x = 11\).

Step by step solution

01

Add 3 to Both Sides of the Equation

Start by adding 3 to both sides of the equation to negate the -3 on the left-hand side. This results in: \(6x - 3 + 3 = 63 + 3\)which can be simplified to: \(6x = 66\)
02

Divide by 6 on Both Sides of the Equation

In order to isolate 'x', divide each side of the equation by 6. This process allows us to find the value of x that makes the equation true. \(6x / 6 = 66 / 6\)which simplifies to: \(x = 11\)
03

Check the Solution

Substitute the solved value of x (11) back into the original equation to verify the solution. \(6*11 - 3 = 63\)This simplifies to: \(63 = 63\)The equation is true, verifying the solution is correct.

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Key Concepts

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

Solving Linear Equations
Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. Solving these equations involves finding the value of the variable that satisfies the equality sign.

To solve a linear equation like \(6x - 3 = 63\), follow these steps:
  • Eliminate constants: Add or subtract to move any constants (numbers without variables) to the other side of the equation. Here, adding 3 to both sides gets rid of the -3, resulting in \(6x = 66\).

  • Isolate the variable: Next, perform operations to get the variable by itself on one side. For our equation, divide both sides by 6 to solve for \(x\), yielding \(x = 11\). This means \(x = 11\) is our solution.
Understanding how each operation affects the equation helps in solving more complex problems methodically. Always aim for one action per step to ensure clarity.
Equation Verification
Equation verification is the process of checking whether the solution to an equation is correct. After solving a linear equation, substitute the variable's value back into the original equation to ensure both sides equalize.

For instance, substituting \(x = 11\) back into the equation \(6x - 3 = 63\), calculate the left side:
  • Replace \(x\) with 11, we have \(6 \times 11 - 3\).
  • Perform the multiplication: \(66 - 3\).
  • Simplify: \(63\).
Finally, comparing both sides confirms they are equal, \(63 = 63\), verifying the solution is correct.

Verification not only checks accuracy but also builds confidence in the problem-solving process. Always perform this step as a final check when solving equations.
Isolation of Variables
Isolation of variables is a key method in solving equations that involves rearranging equations so the unknown variable stands alone on one side of the equation, usually the left.

In the equation \(6x - 3 = 63\), isolating \(x\) involved two main steps:
  • Eliminating constants: Adding 3 to both sides removed the constant on the left, simplifying the equation to \(6x = 66\).
  • Dividing by the coefficient: With the equation \(6x = 66\), divide both sides by 6. This isolates \(x\) completely, resulting in \(x = 11\).
Effective variable isolation helps simplify calculations and is essential when dealing with complex equations. Make sure each operation is justified and mathematically sound, and remember, what you do to one side of the equation, do to the other.

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Most popular questions from this chapter

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{3 x+5}{6-2 x} \geq 0 $$

In Exercises \(1-8,\) add or subtract as indicated and write the result in standard form. $$(-2+6 i)+(4-i)$$

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$\frac{-6-\sqrt{-12}}{48}$$

Solve each quadratic inequality in Exercises \(1-28\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{2}-4 x \geq 0 $$

In Exercises \(9-20,\) find each product and write the result in standard form. $$(8-4 i)(-3+9 i)$$
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