Sure! Let's solve this step-by-step.
We need to find two numbers such that their sum is 12. Let's call the smaller number [tex]\( x \)[/tex] and the larger number [tex]\( y \)[/tex].
1. Setting up the equations:
- From the problem statement, we know:
[tex]\[ x + y = 12 \][/tex]
- We also know that three times the smaller number plus six is seven less than four times the larger number:
[tex]\[ 3x + 6 = 4y - 7 \][/tex]
2. Solving the system of equations:
- Let's start with the first equation:
[tex]\[ x + y = 12 \][/tex]
We can solve this equation for [tex]\( y \)[/tex]:
[tex]\[ y = 12 - x \][/tex]
- Next, substitute [tex]\( y = 12 - x \)[/tex] into the second equation [tex]\( 3x + 6 = 4y - 7 \)[/tex]:
[tex]\[ 3x + 6 = 4(12 - x) - 7 \][/tex]
3. Simplifying the equation:
- Distribute the 4 in the equation:
[tex]\[ 3x + 6 = 48 - 4x - 7 \][/tex]
- Combine like terms on the right side:
[tex]\[ 3x + 6 = 41 - 4x \][/tex]
4. Solving for [tex]\( x \)[/tex]:
- Add [tex]\( 4x \)[/tex] to both sides to get all [tex]\( x \)[/tex] terms on one side:
[tex]\[ 3x + 4x + 6 = 41 \][/tex]
[tex]\[ 7x + 6 = 41 \][/tex]
- Subtract 6 from both sides:
[tex]\[ 7x = 35 \][/tex]
- Divide by 7:
[tex]\[ x = 5 \][/tex]
5. Finding [tex]\( y \)[/tex]:
- Substitute [tex]\( x = 5 \)[/tex] back into the equation [tex]\( y = 12 - x \)[/tex]:
[tex]\[ y = 12 - 5 \][/tex]
[tex]\[ y = 7 \][/tex]
So, the smaller number is 5 and the larger number is 7.
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Example Question Similar to the Given Problem:
The sum of two numbers is 15. The total of four times the smaller number and five is six less than the product of three and the larger. Find the numbers.
1. Setting Up the Equations:
- Let [tex]\( x \)[/tex] be the smaller number and [tex]\( y \)[/tex] be the larger number:
[tex]\[ x + y = 15 \][/tex]
- Four times the smaller number plus five is six less than three times the larger number:
[tex]\[ 4x + 5 = 3y - 6 \][/tex]
2. Solving the System of Equations:
- Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = 15 - x \][/tex]
- Substitute [tex]\( y = 15 - x \)[/tex] into the second equation:
[tex]\[ 4x + 5 = 3(15 - x) - 6 \][/tex]
3. Simplifying the Equation:
- Distribute the 3:
[tex]\[ 4x + 5 = 45 - 3x - 6 \][/tex]
- Combine like terms on the right side:
[tex]\[ 4x + 5 = 39 - 3x \][/tex]
4. Solving for [tex]\( x \)[/tex]:
- Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[ 4x + 3x + 5 = 39 \][/tex]
[tex]\[ 7x + 5 = 39 \][/tex]
- Subtract 5 from both sides:
[tex]\[ 7x = 34 \][/tex]
- Divide by 7:
[tex]\[ x = \frac{34}{7} \][/tex]
[tex]\[ x = 4.857 \][/tex] (approximately)
5. Finding [tex]\( y \)[/tex]:
- Substitute [tex]\( x = \frac{34}{7} \)[/tex] back into the equation [tex]\( y = 15 - x \)[/tex]:
[tex]\[ y = 15 - \frac{34}{7} \][/tex]
[tex]\[ y = \frac{105}{7} - \frac{34}{7} \][/tex]
[tex]\[ y = \frac{71}{7} \][/tex]
[tex]\[ y = 10.143 \][/tex] (approximately)
So, the smaller number is approximately 4.857 and the larger number is approximately 10.143.
