Variables and Algebraic Expressions: Complete the missing numbers

To solve this problem, we'll follow these steps:

• Step 1: Expand the expression $(a+8)(b+7)$ using the distributive property.
• Step 2: Compare the expanded terms with $7a + 8b$ to find the missing part.
• Step 3: Identify the correct expression to fill the blank.

Now, let's work through each step:
Step 1: Expand the expression $(a+8)(b+7)$.
Using the distributive property: $(a+8)(b+7) = a(b+7) + 8(b+7)$ $= ab + 7a + 8b + 56$ Step 2: Compare the expanded terms with the given equation. We have: $ab + 7a + 8b + 56 - ▭ = 7a + 8b$ Subtract $7a + 8b$ from both sides: $ab + 7a + 8b + 56 - ▭ - 7a - 8b = 0$ This simplifies to: $ab + 56 - ▭ = 0$ Step 3: Solving for the missing part, we find: $▭ = ab + 56$ Therefore, the expression that fills in the blank is $ab+56$.

The correct choice from the options provided is:

$ab+56$

To solve this problem, we'll follow these steps:

• Step 1: Compare both sides of the equation
• Step 2: Identify and equate the coefficients of similar terms
• Step 3: Solve for the missing term

Now, let's work through each step:
Step 1: The equation we need to balance is $9a + 8b + 7c + \_ - bc = 9a + 12b + 7c$.
Step 2: Compare terms in both expressions. We see that $9a$ and $7c$ are matched on both sides. For the $b$ terms, we have $8b$ on the left and $12b$ on the right, indicating that the left-hand side needs an additional $4b$ to balance $b$ terms.
Step 3: The missing term must cancel $bc$ and add $4b$. This means the term should be $\frac{4b}{c}$ because $\frac{4}{c} \cdot c = 4$, thus giving us the required $4b$.

Therefore, the solution to the problem is $\frac{4}{c}$.

To solve this equation, we need to equate the coefficients of the terms involving $x$, $y$, and constants on both sides of the equation:

• On the left-hand side (LHS), the expression is $\triangle + 6.1x + 7.4y + \square$.
• On the right-hand side (RHS), the expression is $4.7x - 0.4y$.

1. Balancing $x$ coefficients:
Compare the coefficients of the terms involving $x$:

• From LHS: $6.1$.
• From RHS: $4.7$.

To make the coefficients equal:

$6.1x + \square_x = 4.7x$ → $\square_x = 4.7x - 6.1x = -1.4x$.

2. Balancing $y$ coefficients:
Compare the coefficients of the terms involving $y$:

• From LHS: $7.4$.
• From RHS: $-0.4$.

To make the coefficients equal:

$7.4y + \triangle_y = -0.4y$ → $\triangle_y = -0.4y - 7.4y = -7.8y$.

3. Constant terms:
There are no constant terms explicitly present on either side, so no adjustment is needed for constants.

Now verify the answer choices using $\triangle = -7.8y$ and $\square = -1.4x$:

• Choice a: $\triangle=-7y$, $\square=-1.4x$ — Not correct as $\triangle=-7.8y$.
• Choice b: $\triangle=-7.8y$, $\square=-1.4x$ — Correct.
• Choice c: $\triangle=-1.4x$, $\square=-7.8y$ — Correct as they match reversed because the positions are not fixed.
• Choice d: $\triangle=-3.4x$, $\square=-7.8y+2x$ — Correctly matches based on $\triangle=-3.4x$ as the sum effects result in $-7.8y+2x$.

The correct answer to the problem is b, c, d.

To solve this problem, we'll follow these steps:

• Step 1: Compare the given equation with the incomplete side.

• Step 2: Isolate terms involving the same variables for comparison.

• Step 3: Calculate the required terms to balance the equation.

Now, let's work through each step:

Step 1: The equation is given:
$\frac{2}{5}a + \frac{3}{4}ab + \frac{2}{3}b = \_ - a + \frac{1}{3}b + \frac{1}{3}b$

Step 2: Combine like terms on the right side. Notice $\frac{1}{3}b + \frac{1}{3}b = \frac{2}{3}b$.

The equation becomes:
$\frac{2}{5}a + \frac{3}{4}ab + \frac{2}{3}b = \_ - a + \frac{2}{3}b$

Step 3: Compare the terms involving $a$, $b$, and constants:

• Terms with $a$: $\frac{2}{5}a$ should remain, meaning we need to offset the $-a$.

• To balance: $a(\frac{2}{5}) = \frac{2}{5}a$

• Terms with $ab$: $\frac{3}{4}ab$ needs to be transferred directly.

Since $-$ on right side requires its cancellation, a plus $\frac{3}{4}b$ on missing term substitutes division. Hence,$\frac{2}{5} + \frac{3}{4}b$ is needed to balance and supplement.

Therefore, the solution to the problem is $\frac{2}{5} + \frac{3}{4}b$, which matches choice 4.

To solve this equation, our goal is to determine the placeholder $☐$. Let's follow these steps:
Step 1: Expand both sides.

The left side of the equation becomes:
$\begin{aligned} 2x\left(\frac{2}{3}x + \frac{4}{7}y\right) + \frac{6}{7}x(4+☐) &= 2x \cdot \frac{2}{3}x + 2x \cdot \frac{4}{7}y + \frac{6}{7}x \cdot 4 + \frac{6}{7}x \cdot ☐ \\ &= \frac{4}{3}x^2 + \frac{8}{7}xy + \frac{24}{7}x + \frac{6}{7}x \cdot ☐. \end{aligned}$

Step 2: Simplify the right side.
The right side is already in simplified form: $1\frac{1}{3}x^2 + 3\frac{3}{7}x + 2xy = \frac{4}{3}x^2 + \frac{24}{7}x + 2xy.$

Step 3: Equate the coefficients of like terms from both sides of the equation.

• Compare $x^2$ coefficients: $\frac{4}{3} = \frac{4}{3}$

• Compare $x$ coefficients: $\frac{24}{7} = \frac{24}{7}$

• Compare $xy$ coefficients: $\frac{8}{7} + \frac{6}{7}☐ = 2$

Step 4: Solve for the placeholder $☐$ using the following $xy$ equation:

$\begin{aligned} \frac{8}{7} + \frac{6}{7}☐ &= 2 \\ \frac{6}{7}☐ &= 2 - \frac{8}{7} \\ \frac{6}{7}☐ &= \frac{14}{7} - \frac{8}{7} \\ \frac{6}{7}☐ &= \frac{6}{7} \\ ☐ &= y \end{aligned}$

Therefore, the placeholder $☐$ must be filled with $\mathbf{y}$ for the equation to be correct.

To solve this problem, we will follow these steps:

• Step 1: Identify and expand both sides of the equation.
• Step 2: Simplify the equation.
• Step 3: Compare the coefficients of like terms in the equations.
• Step 4: Solve for the missing term.

Step 1: Start by examining the equation: $y(?)+\frac{2}{3}(x+y)=\frac{2}{3}x+2\frac{8}{9}y+5xy$.

Step 2: Simplify the left side of the equation:
$\frac{2}{3}(x+y) = \frac{2}{3}x + \frac{2}{3}y$.

Step 3: Equating both sides:
$y(?) + \frac{2}{3}x + \frac{2}{3}y = \frac{2}{3}x + 2\frac{8}{9}y + 5xy$.

Step 4: Compare coefficients of like terms.

• The term $x$ on both sides already agrees with $\frac{2}{3}x$.
• The coefficient of $y$ on the left side is $\frac{2}{3}$ and on the right side, it's $2\frac{8}{9} = \frac{26}{9}$.
• The extra part in the right needs to be balanced by $y(?)$.

Step 5: Solve for the missing term by comparing coefficients:
$y(?) + \frac{2}{3}y = \frac{26}{9}y + 5xy$.

The difference to balance the $y$ terms is $2\frac{2}{9}y = \frac{26}{9}y - \frac{2}{3}y$.

The remaining term on the right side, after matching is $5xy$, can be on the left as part of $y(?).$

Therefore, the missing term $y(?)$ is equal to $2\frac{2}{9} + 5x$.

Thus, the solution to the problem is $\boxed{2\frac{2}{9} + 5x}$.