Math / Algebra

Question4 5 $\checkmark 6$ 7 8 9 10 11 12 < 13 14 15 Español 16
The difference of two positive numbers is 7 and the difference of their squares is 147 . Find the numbers. If there is more than one pair, use the "or" button. Give the answer in simplest form.
The pair(s) of numbers: $\square$ and $\square$ $\square \because \square$

Studdy Solution

STEP 1

1. We are given two positive numbers, let's call them $a$ and $b$ .
2. The difference between these numbers is 7: $a - b = 7$ .
3. The difference of their squares is 147: $a^2 - b^2 = 147$ .
4. We need to find the values of $a$ and $b$ .

STEP 2

1. Use the identity for the difference of squares.
2. Substitute the given difference of the numbers.
3. Solve the resulting system of equations.
4. Verify and simplify the solutions.

STEP 3

Recall the identity for the difference of squares:
$a^2 - b^2 = (a - b)(a + b)$

STEP 4

Substitute the given difference $a - b = 7$ into the equation:
$a^2 - b^2 = (a - b)(a + b)$ $147 = 7(a + b)$

STEP 5

Solve for $a + b$ :
$147 = 7(a + b)$ $a + b = \frac{147}{7}$ $a + b = 21$

STEP 6

Now we have a system of linear equations:
1. $a - b = 7$
2. $a + b = 21$
Add these two equations to eliminate $b$ :
$(a - b) + (a + b) = 7 + 21$ $2a = 28$ $a = 14$

STEP 7

Substitute $a = 14$ back into one of the original equations to find $b$ :
$a + b = 21$ $14 + b = 21$ $b = 21 - 14$ $b = 7$

STEP 8

Verify the solution by checking the conditions:
1. $a - b = 14 - 7 = 7$ (Correct)
2. $a^2 - b^2 = 14^2 - 7^2 = 196 - 49 = 147$ (Correct)
The pair of numbers is: $\boxed{14}$ and $\boxed{7}$ .

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Studdy Solution

STEP 1

1. We are given two positive numbers, let's call them $a$ and $b$ .
2. The difference between these numbers is 7: $a - b = 7$ .
3. The difference of their squares is 147: $a^2 - b^2 = 147$ .
4. We need to find the values of $a$ and $b$ .

STEP 2

1. Use the identity for the difference of squares.
2. Substitute the given difference of the numbers.
3. Solve the resulting system of equations.
4. Verify and simplify the solutions.

STEP 3

Recall the identity for the difference of squares:
$a^2 - b^2 = (a - b)(a + b)$

STEP 4

Substitute the given difference $a - b = 7$ into the equation:
$a^2 - b^2 = (a - b)(a + b)$ $147 = 7(a + b)$

STEP 5

Solve for $a + b$ :
$147 = 7(a + b)$ $a + b = \frac{147}{7}$ $a + b = 21$

STEP 6

Now we have a system of linear equations:
1. $a - b = 7$
2. $a + b = 21$
Add these two equations to eliminate $b$ :
$(a - b) + (a + b) = 7 + 21$ $2a = 28$ $a = 14$

STEP 7

Substitute $a = 14$ back into one of the original equations to find $b$ :
$a + b = 21$ $14 + b = 21$ $b = 21 - 14$ $b = 7$

STEP 8

Verify the solution by checking the conditions:
1. $a - b = 14 - 7 = 7$ (Correct)
2. $a^2 - b^2 = 14^2 - 7^2 = 196 - 49 = 147$ (Correct)
The pair of numbers is: $\boxed{14}$ and $\boxed{7}$ .

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Studdy Solution

STEP 1

1. We are given two positive numbers, let's call them a a a and b b b.2. The difference between these numbers is 7: a−b=7 a - b = 7 a−b=7.3. The difference of their squares is 147: a2−b2=147 a^2 - b^2 = 147 a2−b2=147.4. We need to find the values of a a a and b b b.

STEP 2

1. Use the identity for the difference of squares.2. Substitute the given difference of the numbers.3. Solve the resulting system of equations.4. Verify and simplify the solutions.

STEP 3

Recall the identity for the difference of squares:a2−b2=(a−b)(a+b) a^2 - b^2 = (a - b)(a + b) a2−b2=(a−b)(a+b)

STEP 4

Substitute the given difference a−b=7 a - b = 7 a−b=7 into the equation:a2−b2=(a−b)(a+b) a^2 - b^2 = (a - b)(a + b) a2−b2=(a−b)(a+b) 147=7(a+b) 147 = 7(a + b) 147=7(a+b)

STEP 5

Solve for a+b a + b a+b:147=7(a+b) 147 = 7(a + b) 147=7(a+b) a+b=1477 a + b = \frac{147}{7} a+b=7147​ a+b=21 a + b = 21 a+b=21

STEP 6

Now we have a system of linear equations:1. a−b=7 a - b = 7 a−b=72. a+b=21 a + b = 21 a+b=21Add these two equations to eliminate b b b:(a−b)+(a+b)=7+21 (a - b) + (a + b) = 7 + 21 (a−b)+(a+b)=7+21 2a=28 2a = 28 2a=28 a=14 a = 14 a=14

STEP 7

Substitute a=14 a = 14 a=14 back into one of the original equations to find b b b:a+b=21 a + b = 21 a+b=21 14+b=21 14 + b = 21 14+b=21 b=21−14 b = 21 - 14 b=21−14 b=7 b = 7 b=7

STEP 8

Verify the solution by checking the conditions:1. a−b=14−7=7 a - b = 14 - 7 = 7 a−b=14−7=7 (Correct)2. a2−b2=142−72=196−49=147 a^2 - b^2 = 14^2 - 7^2 = 196 - 49 = 147 a2−b2=142−72=196−49=147 (Correct) The pair of numbers is: 14 \boxed{14} 14​ and 7 \boxed{7} 7​.

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1. We are given two positive numbers, let's call them $a$ and $b$ .
2. The difference between these numbers is 7: $a - b = 7$ .
3. The difference of their squares is 147: $a^2 - b^2 = 147$ .
4. We need to find the values of $a$ and $b$ .

1. Use the identity for the difference of squares.
2. Substitute the given difference of the numbers.
3. Solve the resulting system of equations.
4. Verify and simplify the solutions.

Recall the identity for the difference of squares:
$a^2 - b^2 = (a - b)(a + b)$

Substitute the given difference $a - b = 7$ into the equation:
$a^2 - b^2 = (a - b)(a + b)$ $147 = 7(a + b)$

Solve for $a + b$ :
$147 = 7(a + b)$ $a + b = \frac{147}{7}$ $a + b = 21$

Now we have a system of linear equations:
1. $a - b = 7$
2. $a + b = 21$
Add these two equations to eliminate $b$ :
$(a - b) + (a + b) = 7 + 21$ $2a = 28$ $a = 14$

Substitute $a = 14$ back into one of the original equations to find $b$ :
$a + b = 21$ $14 + b = 21$ $b = 21 - 14$ $b = 7$

Verify the solution by checking the conditions:
1. $a - b = 14 - 7 = 7$ (Correct)
2. $a^2 - b^2 = 14^2 - 7^2 = 196 - 49 = 147$ (Correct)
The pair of numbers is: $\boxed{14}$ and $\boxed{7}$ .