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Answer :
Let's tackle each equation one at a time by reasoning about numbers, operations, and expressions:
a. Equation: [tex]\( x + 57 + 94 = 98 + 57 \)[/tex]
1. Notice that the number 57 is present on both sides of the equation. We can think of this as balancing a scale—removing the same weight (57) from both sides will still keep it balanced.
2. When you remove (or subtract) 57 from both sides, you get [tex]\( x + 94 = 98 \)[/tex].
3. Next, think about what number you must add to 94 to get to 98. The answer is 4 because 94 + 4 equals 98.
4. So, [tex]\( x = 4 \)[/tex].
b. Equation: [tex]\( 57 \cdot (94 + x) = 98 \cdot 57 \)[/tex]
1. We see that 57 is a common factor on both sides. This means the expression can be simplified by dividing both sides by 57.
2. After dividing, the equation simplifies to [tex]\( 94 + x = 98 \)[/tex].
3. Again, consider what you need to add to 94 to make it 98. The answer is 4 because 94 + 4 equals 98.
4. Hence, [tex]\( x = 4 \)[/tex].
c. Equation: [tex]\( 57 \cdot 94 + x = 58 \cdot 94 + 3 \)[/tex]
1. Notice that 57 times 94 is on the left and 58 times 94 is on the right. You can think of this as having 57 "bags" each weighing 94 on one side and 58 "bags" weighing the same amount on the other side.
2. By "removing" the impact of 57 bags weighing 94 from both sides, you're left with a simpler problem: just compare the extra amount on each side.
3. The equation simplifies to finding out how one extra 94 (from the 58 instead of 57) on the right compares to the left. The difference is 94, plus the additional 3 on the right side.
4. Therefore, adding 94 and 3 gives you the answer: [tex]\( x = 97 \)[/tex].
These explanations help to understand the logic without relying on standard algebraic methods.
a. Equation: [tex]\( x + 57 + 94 = 98 + 57 \)[/tex]
1. Notice that the number 57 is present on both sides of the equation. We can think of this as balancing a scale—removing the same weight (57) from both sides will still keep it balanced.
2. When you remove (or subtract) 57 from both sides, you get [tex]\( x + 94 = 98 \)[/tex].
3. Next, think about what number you must add to 94 to get to 98. The answer is 4 because 94 + 4 equals 98.
4. So, [tex]\( x = 4 \)[/tex].
b. Equation: [tex]\( 57 \cdot (94 + x) = 98 \cdot 57 \)[/tex]
1. We see that 57 is a common factor on both sides. This means the expression can be simplified by dividing both sides by 57.
2. After dividing, the equation simplifies to [tex]\( 94 + x = 98 \)[/tex].
3. Again, consider what you need to add to 94 to make it 98. The answer is 4 because 94 + 4 equals 98.
4. Hence, [tex]\( x = 4 \)[/tex].
c. Equation: [tex]\( 57 \cdot 94 + x = 58 \cdot 94 + 3 \)[/tex]
1. Notice that 57 times 94 is on the left and 58 times 94 is on the right. You can think of this as having 57 "bags" each weighing 94 on one side and 58 "bags" weighing the same amount on the other side.
2. By "removing" the impact of 57 bags weighing 94 from both sides, you're left with a simpler problem: just compare the extra amount on each side.
3. The equation simplifies to finding out how one extra 94 (from the 58 instead of 57) on the right compares to the left. The difference is 94, plus the additional 3 on the right side.
4. Therefore, adding 94 and 3 gives you the answer: [tex]\( x = 97 \)[/tex].
These explanations help to understand the logic without relying on standard algebraic methods.
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