Solving a System of Equations for Three Numbers with a Sum of 22

Math Problem Statement

Find three numbers such that their sum is 22 and the second is 5 more than the first and 2 more than the third.

Solution

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The problem in the image appears to involve solving a system of linear equations. The text reads:

Find three numbers such that their sum is 22 and the second is 5 more than the first and 2 more than the third.

To solve this, we can define three variables:

Let the first number be $x$ .
Let the second number be $y$ .
Let the third number be $z$ .

Given:

$x + y + z = 22$ (The sum of the three numbers is 22)
$y = x + 5$ (The second number is 5 more than the first)
$y = z + 2$ (The second number is 2 more than the third)

Now, we have a system of equations:

1. & \quad x + y + z = 22 \\ 2. & \quad y = x + 5 \\ 3. & \quad y = z + 2 \end{aligned}$$ ### Solution 1. From equation 2: $$y = x + 5$$. 2. From equation 3: $$y = z + 2$$. Substituting $$y = x + 5$$ into $$y = z + 2$$: $$x + 5 = z + 2$$ This simplifies to: $$z = x + 3$$ Now substitute $$y = x + 5$$ and $$z = x + 3$$ into the first equation: $$x + (x + 5) + (x + 3) = 22$$ This simplifies to: $$3x + 8 = 22$$ Subtract 8 from both sides: $$3x = 14$$ Divide by 3: $$x = \frac{14}{3} \approx 4.67$$ Now substitute $$x$$ back into the equations for $$y$$ and $$z$$: $$y = x + 5 = \frac{14}{3} + 5 \approx 9.67$$ $$z = x + 3 = \frac{14}{3} + 3 \approx 7.67$$ So, the three numbers are approximately: - First number: $$x = \frac{14}{3}$$ - Second number: $$y = \frac{29}{3}$$ - Third number: $$z = \frac{23}{3}$$ These fractions simplify to: - First number: $$x \approx 4.67$$ - Second number: $$y \approx 9.67$$ - Third number: $$z \approx 7.67$$ Would you like further details or clarification on this problem? --- ### Related Questions: 1. How would the solution change if the second number was 4 more than the first instead of 5? 2. Can you solve a similar problem with different conditions, such as different sums? 3. What if the problem involved four numbers instead of three? 4. How does the method change if the relationships between the numbers are multiplicative rather than additive? 5. Can this problem be solved using a matrix or linear algebra method? --- ### Tip: When solving systems of equations, always double-check the relationships defined in the problem statement to ensure correct substitution and simplification.

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Math Problem Analysis

Mathematical Concepts

Algebra
System of Linear Equations

Formulas

x + y + z = 22
y = x + 5
y = z + 2
z = x + 3
3x + 8 = 22

Theorems

Suitable Grade Level

Grades 8-10

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