Daniel is four years older than Greg. In 8 years, Daniel will be two years older than three times Greg's current age. How old is Greg now?

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Let's solve this age problem. Daniel is four years older than Greg. In 8 years, Daniel will be two years older than three times Greg's current age. We need to find Greg's current age. Let's define G as Greg's current age and D as Daniel's current age. From the first statement, we know that Daniel is 4 years older than Greg. Now, let's set up equations based on the given information. From the first statement, we know that Daniel is 4 years older than Greg, so D equals G plus 4. For the second statement, in 8 years, Daniel's age will be D plus 8. This will be equal to 3 times Greg's current age plus 2 years. So our second equation is D plus 8 equals 3G plus 2. Now we have two equations with two unknowns, which we can solve to find Greg's current age. Now, let's solve these equations to find Greg's age. We have D equals G plus 4, and D plus 8 equals 3G plus 2. First, we substitute the first equation into the second. This gives us G plus 4 plus 8 equals 3G plus 2, which simplifies to G plus 12 equals 3G plus 2. Moving all G terms to the left side, we get G minus 3G plus 12 equals 2, which is negative 2G plus 12 equals 2. Subtracting 12 from both sides gives us negative 2G equals negative 10. Dividing both sides by negative 2, we get G equals 5. Therefore, Greg is currently 5 years old. We can verify this: if Greg is 5, then Daniel is 9. In 8 years, Daniel will be 17, which is indeed 2 more than 3 times Greg's current age, which is 15. Let's summarize what we've done to solve this problem. We were asked to find Greg's current age based on two conditions: Daniel is four years older than Greg, and in 8 years, Daniel will be two years older than three times Greg's current age. We defined variables G for Greg's age and D for Daniel's age. Then we set up two equations: D equals G plus 4, and D plus 8 equals 3G plus 2. By substituting the first equation into the second and solving algebraically, we found that Greg is currently 5 years old, which means Daniel is 9 years old. We verified our answer by checking that in 8 years, Daniel will be 17, which is indeed 2 more than 3 times Greg's current age of 5. Let's review the key takeaways from solving this age problem. When approaching word problems like this, it's important to follow a systematic strategy. First, define your variables clearly - in our case, G for Greg's age and D for Daniel's age. Second, carefully translate the word problem into mathematical equations. We got D equals G plus 4, and D plus 8 equals 3G plus 2. Third, use algebraic techniques like substitution to solve the system of equations. Fourth, always verify your solution by checking it against the original conditions. And finally, interpret your answer in the context of the problem - Greg is 5 years old and Daniel is 9 years old. The graph shows our two equations intersecting at the point (5, 9), confirming our algebraic solution. This systematic approach can be applied to many different types of word problems.