Answer :
Sure! Let's solve each of these equations step by step by reasoning about numbers, operations, and expressions.
a. Equation: [tex]\( x + 57 + 94 = 98 + 57 \)[/tex]
- Combine like terms on both sides:
- Left side: [tex]\( x + (57 + 94) = x + 151 \)[/tex]
- Right side: [tex]\( 98 + 57 = 155 \)[/tex]
- The equation becomes: [tex]\( x + 151 = 155 \)[/tex]
- To find [tex]\( x \)[/tex], subtract 151 from both sides:
- [tex]\( x = 155 - 151 = 4 \)[/tex]
b. Equation: [tex]\( 57 \cdot (94 + x) = 98 \cdot 57 \)[/tex]
- Simplify by distributing 57:
- Left side: [tex]\( 57 \cdot 94 + 57 \cdot x = 5358 + 57x \)[/tex]
- Right side: Calculate [tex]\( 98 \cdot 57 = 5586 \)[/tex]
- The equation becomes: [tex]\( 5358 + 57x = 5586 \)[/tex]
- Subtract 5358 from both sides:
- [tex]\( 57x = 5586 - 5358 = 228 \)[/tex]
- Divide both sides by 57 to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 228 / 57 = 4 \)[/tex]
c. Equation: [tex]\( 57 \cdot 94 + x = 58 \cdot 94 + 3 \)[/tex]
- Simplify both sides:
- Left side: [tex]\( 57 \cdot 94 = 5358 \)[/tex] so [tex]\( 5358 + x \)[/tex]
- Right side: [tex]\( 58 \cdot 94 = 5452 \)[/tex] so [tex]\( 5452 + 3 = 5455 \)[/tex]
- The equation becomes: [tex]\( 5358 + x = 5455 \)[/tex]
- Subtract 5358 from both sides to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 5455 - 5358 = 97 \)[/tex]
d. Equation: [tex]\( 487 + 176 = x + 490 \)[/tex]
- Combine like terms:
- Left side: [tex]\( 487 + 176 = 663 \)[/tex]
- The equation becomes: [tex]\( 663 = x + 490 \)[/tex]
- Subtract 490 from both sides:
- [tex]\( x = 663 - 490 = 173 \)[/tex]
e. Equation: [tex]\( 333 \cdot 213 = 111 \cdot A \)[/tex]
- First, calculate the left side:
- [tex]\( 333 \times 213 = 70889 \)[/tex]
- Now, divide both sides by 111 to solve for [tex]\( A \)[/tex]:
- [tex]\( A = 70889 / 111 = 639 \)[/tex]
f. Equation: [tex]\( C + 34 \cdot 8 = 28 \cdot 34 \)[/tex]
- Calculate both sides:
- [tex]\( 34 \cdot 8 = 272 \)[/tex]
- [tex]\( 28 \cdot 34 = 952 \)[/tex]
- The equation becomes: [tex]\( C + 272 = 952 \)[/tex]
- Subtract 272 to solve for [tex]\( C \)[/tex]:
- [tex]\( C = 952 - 272 = 680 \)[/tex]
g. Equation: [tex]\( \frac{1}{4} x + \frac{7}{16} = 11 \frac{7}{16} \)[/tex]
- Write [tex]\( 11 \frac{7}{16} \)[/tex] as an improper fraction: [tex]\( = \frac{183}{16} \)[/tex]
- Subtract [tex]\( \frac{7}{16} \)[/tex] from both sides:
- [tex]\( \frac{1}{4} x = \frac{183}{16} - \frac{7}{16} = \frac{176}{16} = 11 \)[/tex]
- Multiply both sides by 4 to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 11 \cdot 4 = 44 \)[/tex]
h. Equation: [tex]\( 42 = x - 42 \)[/tex]
- Add 42 to both sides to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 42 + 42 = 84 \)[/tex]
These step-by-step solutions provide a clear understanding of how each equation is approached by reasoning through numbers and operations.
a. Equation: [tex]\( x + 57 + 94 = 98 + 57 \)[/tex]
- Combine like terms on both sides:
- Left side: [tex]\( x + (57 + 94) = x + 151 \)[/tex]
- Right side: [tex]\( 98 + 57 = 155 \)[/tex]
- The equation becomes: [tex]\( x + 151 = 155 \)[/tex]
- To find [tex]\( x \)[/tex], subtract 151 from both sides:
- [tex]\( x = 155 - 151 = 4 \)[/tex]
b. Equation: [tex]\( 57 \cdot (94 + x) = 98 \cdot 57 \)[/tex]
- Simplify by distributing 57:
- Left side: [tex]\( 57 \cdot 94 + 57 \cdot x = 5358 + 57x \)[/tex]
- Right side: Calculate [tex]\( 98 \cdot 57 = 5586 \)[/tex]
- The equation becomes: [tex]\( 5358 + 57x = 5586 \)[/tex]
- Subtract 5358 from both sides:
- [tex]\( 57x = 5586 - 5358 = 228 \)[/tex]
- Divide both sides by 57 to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 228 / 57 = 4 \)[/tex]
c. Equation: [tex]\( 57 \cdot 94 + x = 58 \cdot 94 + 3 \)[/tex]
- Simplify both sides:
- Left side: [tex]\( 57 \cdot 94 = 5358 \)[/tex] so [tex]\( 5358 + x \)[/tex]
- Right side: [tex]\( 58 \cdot 94 = 5452 \)[/tex] so [tex]\( 5452 + 3 = 5455 \)[/tex]
- The equation becomes: [tex]\( 5358 + x = 5455 \)[/tex]
- Subtract 5358 from both sides to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 5455 - 5358 = 97 \)[/tex]
d. Equation: [tex]\( 487 + 176 = x + 490 \)[/tex]
- Combine like terms:
- Left side: [tex]\( 487 + 176 = 663 \)[/tex]
- The equation becomes: [tex]\( 663 = x + 490 \)[/tex]
- Subtract 490 from both sides:
- [tex]\( x = 663 - 490 = 173 \)[/tex]
e. Equation: [tex]\( 333 \cdot 213 = 111 \cdot A \)[/tex]
- First, calculate the left side:
- [tex]\( 333 \times 213 = 70889 \)[/tex]
- Now, divide both sides by 111 to solve for [tex]\( A \)[/tex]:
- [tex]\( A = 70889 / 111 = 639 \)[/tex]
f. Equation: [tex]\( C + 34 \cdot 8 = 28 \cdot 34 \)[/tex]
- Calculate both sides:
- [tex]\( 34 \cdot 8 = 272 \)[/tex]
- [tex]\( 28 \cdot 34 = 952 \)[/tex]
- The equation becomes: [tex]\( C + 272 = 952 \)[/tex]
- Subtract 272 to solve for [tex]\( C \)[/tex]:
- [tex]\( C = 952 - 272 = 680 \)[/tex]
g. Equation: [tex]\( \frac{1}{4} x + \frac{7}{16} = 11 \frac{7}{16} \)[/tex]
- Write [tex]\( 11 \frac{7}{16} \)[/tex] as an improper fraction: [tex]\( = \frac{183}{16} \)[/tex]
- Subtract [tex]\( \frac{7}{16} \)[/tex] from both sides:
- [tex]\( \frac{1}{4} x = \frac{183}{16} - \frac{7}{16} = \frac{176}{16} = 11 \)[/tex]
- Multiply both sides by 4 to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 11 \cdot 4 = 44 \)[/tex]
h. Equation: [tex]\( 42 = x - 42 \)[/tex]
- Add 42 to both sides to solve for [tex]\( x \)[/tex]:
- [tex]\( x = 42 + 42 = 84 \)[/tex]
These step-by-step solutions provide a clear understanding of how each equation is approached by reasoning through numbers and operations.
