We are given the linear equation 3 times the quantity x minus 2 equals 20. Our goal is to solve for the value of x. Let's break this down: we have a coefficient of 3, a variable expression in parentheses (x minus 2), and a constant value of 20 on the right side.
To solve this equation, we first apply the distributive property. This means we multiply the coefficient 3 by each term inside the parentheses. So 3 times x gives us 3x, and 3 times negative 2 gives us negative 6. This transforms our equation to 3x minus 6 equals 20.
Now we want to isolate the term with x. To do this, we use the addition property of equality, which states that we can add the same value to both sides of an equation without changing its solution. We add 6 to both sides: on the left, negative 6 plus 6 cancels out, leaving us with 3x. On the right, 20 plus 6 equals 26. So our equation becomes 3x equals 26.
To solve for x, we need to eliminate the coefficient 3. We do this by dividing both sides of the equation by 3. On the left side, 3x divided by 3 simplifies to x. On the right side, we have 26 divided by 3. This gives us x equals 26 thirds. We can also express this as a mixed number: 8 and 2 thirds, or approximately 8.667 in decimal form.
To verify our solution is correct, we substitute x equals 26 thirds back into the original equation. This gives us 3 times the quantity 26 thirds minus 2 equals 20. First, we convert 2 to a fraction with denominator 3, giving us 6 thirds. Then we subtract: 26 thirds minus 6 thirds equals 20 thirds. Now we multiply: 3 times 20 thirds equals 20. Since both sides of the equation equal 20, our solution is verified.
Let's summarize the complete solution process. First, we started with the original equation 3 times the quantity x minus 2 equals 20. Second, we applied the distributive property to get 3x minus 6 equals 20. Third, we added 6 to both sides to isolate the variable term, resulting in 3x equals 26. Fourth, we divided both sides by 3 to solve for x, giving us x equals 26 thirds. Finally, we expressed this as a mixed number: 8 and 2 thirds. Throughout this process, we used key algebraic principles: the distributive property, addition property of equality, and division property of equality. Our solution is verified and correct.