Question

practice & problem solvin leveled practice in 6–9, factor the expression. 6. 16a + 10. the gcf of 16a and 10 is 2. 2 × □ = 16a 2 × □ = 10 the factored expression is □. 8. \\(\frac{1}{4}x + \frac{7}{4}\\) 10. represent and connect this area model shows th write two expressions that represent the area. xxxxx111111

Explanation:

Step1: Solve for the first blank in 6.

We know that \(2\times\square = 16a\). To find the value in the square, we divide both sides of the equation by 2. So, \(\square=\frac{16a}{2} = 8a\).

Step2: Solve for the second blank in 6.

We have \(2\times\square = 10\). Dividing both sides by 2, we get \(\square=\frac{10}{2}=5\).

Step3: Find the factored expression in 6.

Using the distributive property \(ab + ac=a(b + c)\), where \(a = 2\), \(b = 8a\), and \(c = 5\), the factored expression is \(2(8a + 5)\).

Step4: Factor the expression in 8.

For the expression \(\frac{1}{4}x+\frac{7}{4}\), the greatest common factor (GCF) of \(\frac{1}{4}x\) and \(\frac{7}{4}\) is \(\frac{1}{4}\). Using the distributive property, we can factor it as \(\frac{1}{4}(x + 7)\).

Step5: Solve problem 10 (assuming the area model has 5 blue \(x\) tiles and 6 green 1 tiles).

First expression:

The area can be represented as the sum of the areas of the blue tiles and the green tiles. The area of the blue tiles is \(5\times x=5x\) and the area of the green tiles is \(6\times1 = 6\). So, one expression is \(5x+6\).

Second expression:

If we consider the total number of tiles, we can also think of it as a rectangle (assuming) with length \((x + 1)\) but wait, no, actually, since we have 5 \(x\)s and 6 \(1\)s, another way is to factor? Wait, no, the first way is sum, the second way can be if we consider the GCF? Wait, no, the area model here is just adding the two parts. Wait, maybe the two expressions are the expanded form \(5x + 6\) and the factored form? But there is no common factor between 5 and 6. Wait, maybe I misread the tiles. Wait, the blue tiles: 5 tiles with \(x\), green tiles: 6 tiles with 1. So first expression: \(x+x+x+x+x + 1+1+1+1+1+1=5x + 6\). Second expression: if we consider the total area as the sum, but maybe the problem expects \(5x+6\) and \(1\times(5x + 6)\) or maybe another way. Wait, perhaps the two expressions are the sum of the individual areas and the factored form (but since GCF of 5 and 6 is 1, the factored form is \(1(5x + 6)\)) or maybe the problem has a different number of tiles. Wait, maybe I made a mistake. Let's re - examine. The blue tiles: 5 \(x\)s, green tiles: 6 \(1\)s. So first expression: \(5x+6\), second expression: \(x\times5+1\times6\) or \(5(x)+6(1)\) or just \(5x + 6\) and \(1\times(5x + 6)\). But maybe the intended answer is \(5x+6\) and \( (x + 1)\times(5 + 6)\)? No, that's not correct. Wait, maybe the area model is a rectangle with length 5 and width \(x\) plus a rectangle with length 6 and width 1, but that's the same as \(5x+6\). Alternatively, maybe the two expressions are \(5x+6\) and \(x\times5+6\times1\), but that's the same. Wait, perhaps the problem has a different number of tiles. Wait, the user's image shows "x x x x x" (5 x's) and "1 1 1 1 1 1" (6 1's). So:

First expression: \(5x + 6\)

Second expression: We can also write it as \(x+x+x+x+x+1+1+1+1+1+1\) (the expanded form of the sum), but that's not helpful. Wait, maybe the problem expects the two expressions as the sum \(5x + 6\) and the factored form, but since 5 and 6 have no common factor, maybe I made a mistake. Wait, no, maybe the tiles are 5 x's and 6 1's, so the area is \(5x+6\), and another way is to consider the total number of tiles as a single term? No, that doesn't make sense. Alternatively, maybe the area model is a rectangle with length \((x + 1)\) and width 5, but no, because there are 6 ones. Wait, I think I will proceed with the two expressions as \(5x+6\) (sum of the two regions) and if we consider the GCF is 1, then \(1\times(5x + 6)\), but that's trivial. Maybe the problem has a typo or I misread the tiles. Let's assume that the two expressions are \(5x + 6\) and \(x\times5+1\times6\) (but that's the same as the first one). Alternatively, maybe the number of x tiles is 5 and 1 tiles is 6, so the two expressions are \(5x+6\) and \(6 + 5x\) (commutative property), but that's also the same. Wait, maybe the original problem's area model has different tiles. Anyway, proceeding with the above.

Answer:

• For problem 6: The first blank is \(8a\), the second blank is \(5\), and the factored expression is \(2(8a + 5)\).
• For problem 8: The factored expression is \(\frac{1}{4}(x + 7)\).
• For problem 10: The two expressions are \(5x+6\) and \(x+x+x+x+x + 1+1+1+1+1+1\) (or \(5x+6\) and \(1\times(5x + 6)\) but more likely \(5x + 6\) and the sum of the individual tiles as \(x+x+x+x+x+1+1+1+1+1+1\)).