Question

find the error in the system ( a + b = 7 ) and ( 1.29a + 0.49b = 6.63 ), ( a ) represents pounds of apples and ( b ) represents pounds of bananas a person bought. your classmate found and interpreted the solution as shown. ( 1.29a + 0.49b = 6.63 ) ( 1.29a + 0.49(a + 7) = 6.63 ) ( 1.29a + 0.49a + 3.43 = 6.63 ) ( 1.78a = 3.2 ) ( a = 1.8 ) ( a + b = 7 ), so ( b = 5.2 ). the solution ( (1.8, 5.2) ) means that 1.8 pounds of apples and 5.2 pounds of bananas were bought. describe the classmate’s error and the feedback you can provide to help them correct the error. my classmate substituted incorrectly for (\boxed{quad}). they should have substituted (\boxed{quad}) for (\boxed{quad}) to find that the person bought (\boxed{quad}) pounds of apples and (\boxed{quad}) pounds of bananas.

Explanation:

Step1: Identify the error

In the step \(1.29a + 0.49(a + 7)=6.63\), the classmate incorrectly substituted \(b\) with \(a + 7\). But from \(a + b=7\), we know \(b = 7 - a\), not \(a + 7\). So the error is in the substitution of \(b\); they used \(a + 7\) instead of \(7 - a\).

Step2: Correct substitution

We have the system:

1. \(a + b=7\) (so \(b = 7 - a\))
2. \(1.29a+0.49b = 6.63\)

Substitute \(b = 7 - a\) into the second equation:
\(1.29a+0.49(7 - a)=6.63\)
Expand: \(1.29a + 3.43-0.49a=6.63\)
Combine like terms: \((1.29a - 0.49a)+3.43 = 6.63\)
\(0.8a+3.43 = 6.63\)
Subtract 3.43 from both sides: \(0.8a=6.63 - 3.43=3.2\)
Divide by 0.8: \(a=\frac{3.2}{0.8}=4\)
Then from \(a + b=7\), \(b = 7 - 4 = 3\)

So the classmate substituted incorrectly for \(b\) (used \(a + 7\) instead of \(7 - a\)). They should have substituted \(7 - a\) for \(b\) to find that the person bought \(4\) pounds of apples and \(3\) pounds of bananas.

Answer:

My classmate substituted incorrectly for \(\boldsymbol{b}\) (used \(a + 7\) instead of \(7 - a\)). They should have substituted \(\boldsymbol{7 - a}\) for \(\boldsymbol{b}\) to find that the person bought \(\boldsymbol{4}\) pounds of apples and \(\boldsymbol{3}\) pounds of bananas.