Solved on Feb 04, 2024

Find the values to complete the equation $2(\square-4x)+\square(3x-1)=2(2x+3)$ .

STEP 1

Assumptions
1. We are given a linear equation with two boxes that represent unknown numbers.
2. We need to find the values of these unknown numbers that make the equation true.

STEP 2

First, we need to expand the expressions on both sides of the equation.
$2(\square-4x)+\square(3x-1)=2(2x+3)$

STEP 3

Expand the left side of the equation by distributing the multiplication over subtraction for the first term and over subtraction for the second term.
$2 \cdot \square - 2 \cdot 4x + \square \cdot 3x - \square \cdot 1 = 2(2x+3)$

STEP 4

Simplify the left side of the equation by combining like terms.
$2 \cdot \square + (\square \cdot 3x - 8x) - \square = 2(2x+3)$

STEP 5

Now, expand the right side of the equation by distributing the multiplication over addition.
$2 \cdot \square + (\square \cdot 3x - 8x) - \square = 4x + 6$

STEP 6

We now have an equation with like terms on both sides. We can set the coefficients of the like terms equal to each other to solve for the unknown numbers in the boxes.

STEP 7

Let's denote the number in the first box as $a$ and the number in the second box as $b$ . We can rewrite the equation as:
$2a + (b \cdot 3x - 8x) - b = 4x + 6$

STEP 8

Now, we can collect the terms with $x$ on one side and the constant terms on the other side.
$2a - b + (b \cdot 3x - 8x) = 4x + 6$

STEP 9

Combine the terms with $x$ .
$(3b - 8)x = 4x$

STEP 10

Since the coefficients of $x$ must be equal on both sides of the equation, we can equate them.
$3b - 8 = 4$

STEP 11

Solve for $b$ .
$3b = 4 + 8$
$3b = 12$
$b = \frac{12}{3}$
$b = 4$

STEP 12

Now that we have the value of $b$ , we can substitute it back into the equation to find $a$ .
$2a - b = 6$

STEP 13

Substitute $b = 4$ into the equation.
$2a - 4 = 6$

STEP 14

Solve for $a$ .
$2a = 6 + 4$
$2a = 10$
$a = \frac{10}{2}$
$a = 5$

STEP 15

Now we have the values for both boxes. The number in the first box is $5$ and the number in the second box is $4$ .
The completed calculation is:
$2(5-4x)+4(3x-1)=2(2x+3)$

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Find the values to complete the equation $2(\square-4x)+\square(3x-1)=2(2x+3)$ .

STEP 1

Assumptions
1. We are given a linear equation with two boxes that represent unknown numbers.
2. We need to find the values of these unknown numbers that make the equation true.

STEP 2

First, we need to expand the expressions on both sides of the equation.
$2(\square-4x)+\square(3x-1)=2(2x+3)$

STEP 3

Expand the left side of the equation by distributing the multiplication over subtraction for the first term and over subtraction for the second term.
$2 \cdot \square - 2 \cdot 4x + \square \cdot 3x - \square \cdot 1 = 2(2x+3)$

STEP 4

Simplify the left side of the equation by combining like terms.
$2 \cdot \square + (\square \cdot 3x - 8x) - \square = 2(2x+3)$

STEP 5

Now, expand the right side of the equation by distributing the multiplication over addition.
$2 \cdot \square + (\square \cdot 3x - 8x) - \square = 4x + 6$

STEP 6

We now have an equation with like terms on both sides. We can set the coefficients of the like terms equal to each other to solve for the unknown numbers in the boxes.

STEP 7

Let's denote the number in the first box as $a$ and the number in the second box as $b$ . We can rewrite the equation as:
$2a + (b \cdot 3x - 8x) - b = 4x + 6$

STEP 8

Now, we can collect the terms with $x$ on one side and the constant terms on the other side.
$2a - b + (b \cdot 3x - 8x) = 4x + 6$

STEP 9

Combine the terms with $x$ .
$(3b - 8)x = 4x$

STEP 10

Since the coefficients of $x$ must be equal on both sides of the equation, we can equate them.
$3b - 8 = 4$

STEP 11

Solve for $b$ .
$3b = 4 + 8$
$3b = 12$
$b = \frac{12}{3}$
$b = 4$

STEP 12

Now that we have the value of $b$ , we can substitute it back into the equation to find $a$ .
$2a - b = 6$

STEP 13

Substitute $b = 4$ into the equation.
$2a - 4 = 6$

STEP 14

Solve for $a$ .
$2a = 6 + 4$
$2a = 10$
$a = \frac{10}{2}$
$a = 5$

STEP 15

Now we have the values for both boxes. The number in the first box is $5$ and the number in the second box is $4$ .
The completed calculation is:
$2(5-4x)+4(3x-1)=2(2x+3)$

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Find the values to complete the equation 2(□−4x)+□(3x−1)=2(2x+3)2(\square-4x)+\square(3x-1)=2(2x+3)2(□−4x)+□(3x−1)=2(2x+3).

STEP 1

Assumptions1. We are given a linear equation with two boxes that represent unknown numbers.2. We need to find the values of these unknown numbers that make the equation true.

STEP 2

First, we need to expand the expressions on both sides of the equation.2(□−4x)+□(3x−1)=2(2x+3)2(\square-4x)+\square(3x-1)=2(2x+3)2(□−4x)+□(3x−1)=2(2x+3)

STEP 3

STEP 4

Simplify the left side of the equation by combining like terms.2⋅□+(□⋅3x−8x)−□=2(2x+3)2 \cdot \square + (\square \cdot 3x - 8x) - \square = 2(2x+3)2⋅□+(□⋅3x−8x)−□=2(2x+3)

STEP 5

Now, expand the right side of the equation by distributing the multiplication over addition.2⋅□+(□⋅3x−8x)−□=4x+62 \cdot \square + (\square \cdot 3x - 8x) - \square = 4x + 62⋅□+(□⋅3x−8x)−□=4x+6

STEP 6

We now have an equation with like terms on both sides. We can set the coefficients of the like terms equal to each other to solve for the unknown numbers in the boxes.

STEP 7

Let's denote the number in the first box as aaa and the number in the second box as bbb. We can rewrite the equation as:2a+(b⋅3x−8x)−b=4x+62a + (b \cdot 3x - 8x) - b = 4x + 62a+(b⋅3x−8x)−b=4x+6

STEP 8

Now, we can collect the terms with xxx on one side and the constant terms on the other side.2a−b+(b⋅3x−8x)=4x+62a - b + (b \cdot 3x - 8x) = 4x + 62a−b+(b⋅3x−8x)=4x+6

STEP 9

Combine the terms with xxx.(3b−8)x=4x(3b - 8)x = 4x(3b−8)x=4x

STEP 10

Since the coefficients of xxx must be equal on both sides of the equation, we can equate them.3b−8=43b - 8 = 43b−8=4

STEP 11

Solve for bbb.3b=4+83b = 4 + 83b=4+83b=123b = 123b=12b=123b = \frac{12}{3}b=312​b=4b = 4b=4

STEP 12

Now that we have the value of bbb, we can substitute it back into the equation to find aaa.2a−b=62a - b = 62a−b=6

STEP 13

Substitute b=4b = 4b=4 into the equation.2a−4=62a - 4 = 62a−4=6

STEP 14

Solve for aaa.2a=6+42a = 6 + 42a=6+42a=102a = 102a=10a=102a = \frac{10}{2}a=210​a=5a = 5a=5

STEP 15

Now we have the values for both boxes. The number in the first box is 555 and the number in the second box is 444.The completed calculation is:2(5−4x)+4(3x−1)=2(2x+3) 2(5-4x)+4(3x-1)=2(2x+3) 2(5−4x)+4(3x−1)=2(2x+3)

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the values to complete the equation $2(\square-4x)+\square(3x-1)=2(2x+3)$ .

Assumptions
1. We are given a linear equation with two boxes that represent unknown numbers.
2. We need to find the values of these unknown numbers that make the equation true.

First, we need to expand the expressions on both sides of the equation.
$2(\square-4x)+\square(3x-1)=2(2x+3)$

Simplify the left side of the equation by combining like terms.
$2 \cdot \square + (\square \cdot 3x - 8x) - \square = 2(2x+3)$

Now, expand the right side of the equation by distributing the multiplication over addition.
$2 \cdot \square + (\square \cdot 3x - 8x) - \square = 4x + 6$

Let's denote the number in the first box as $a$ and the number in the second box as $b$ . We can rewrite the equation as:
$2a + (b \cdot 3x - 8x) - b = 4x + 6$

Now, we can collect the terms with $x$ on one side and the constant terms on the other side.
$2a - b + (b \cdot 3x - 8x) = 4x + 6$

Combine the terms with $x$ .
$(3b - 8)x = 4x$

Since the coefficients of $x$ must be equal on both sides of the equation, we can equate them.
$3b - 8 = 4$

Solve for $b$ .
$3b = 4 + 8$
$3b = 12$
$b = \frac{12}{3}$
$b = 4$

Now that we have the value of $b$ , we can substitute it back into the equation to find $a$ .
$2a - b = 6$

Substitute $b = 4$ into the equation.
$2a - 4 = 6$

Solve for $a$ .
$2a = 6 + 4$
$2a = 10$
$a = \frac{10}{2}$
$a = 5$

Now we have the values for both boxes. The number in the first box is $5$ and the number in the second box is $4$ .
The completed calculation is:
$2(5-4x)+4(3x-1)=2(2x+3)$