We are given the linear equation 3 times open parenthesis x minus 2 close parenthesis equals 22. Our goal is to find the value of x that makes this equation true. Let's examine the components: we have a coefficient of 3, an expression in parentheses (x minus 2), and a constant 22 on the right side.
To solve this equation, we first apply the distributive property to expand the left side. The distributive property states that a times open parenthesis b plus c close parenthesis equals a times b plus a times c. Applying this to 3 times open parenthesis x minus 2 close parenthesis, we multiply 3 by x to get 3x, and 3 by negative 2 to get negative 6. This transforms our equation to 3x minus 6 equals 22.
Now we want to isolate the term with x. To do this, we use the addition property of equality, which states that adding the same value to both sides of an equation maintains the equality. We add 6 to both sides to eliminate the minus 6 on the left. This gives us 3x minus 6 plus 6 equals 22 plus 6. Simplifying both sides, we get 3x equals 28. The balance scale visualization shows how adding the same value to both sides keeps the equation balanced.
To solve for x, we need to eliminate the coefficient 3. We do this by dividing both sides of the equation by 3, using the division property of equality. This gives us 3x divided by 3 equals 28 divided by 3. Simplifying the left side, 3x divided by 3 is just x. On the right side, 28 divided by 3 is the improper fraction 28 thirds. This can also be expressed as the mixed number 9 and 1 third.
To verify our solution is correct, we substitute x equals 28 thirds back into the original equation 3 times open parenthesis x minus 2 close parenthesis equals 22. This gives us 3 times open parenthesis 28 thirds minus 2 close parenthesis equals 22. First, we convert 2 to thirds: 2 equals 6 thirds. So we have 3 times open parenthesis 28 thirds minus 6 thirds close parenthesis equals 22. Simplifying inside the parentheses: 28 thirds minus 6 thirds equals 22 thirds. Now we have 3 times 22 thirds equals 22. Multiplying: 3 times 22 thirds equals 66 thirds, which simplifies to 22. Since both sides equal 22, our solution is verified.
Let's summarize the complete solution process. First, we started with the original equation: 3 times open parenthesis x minus 2 close parenthesis equals 22. Next, we applied the distributive property to expand the left side, giving us 3x minus 6 equals 22. Then, we used the addition property of equality by adding 6 to both sides, resulting in 3x equals 28. After that, we applied the division property of equality by dividing both sides by 3, which gave us x equals 28 thirds. Finally, we expressed this as the mixed number 9 and 1 third. This systematic approach of expanding, isolating, solving, and verifying is fundamental to solving linear equations.