To find the solution to the given equations and determine the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex], let's detail the process step by step.
### Step 1: Define the Relationships Between the Variables
We start with the initial set of proportional equations:
[tex]\[ \frac{x}{4} = \frac{y}{3} = \frac{z}{2} \][/tex]
From this proportionality, we can express [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] in terms of a common variable [tex]\(k\)[/tex]:
[tex]\[ x = 4k \][/tex]
[tex]\[ y = 3k \][/tex]
[tex]\[ z = 2k \][/tex]
### Step 2: Substitute Into the Second Equation
We substitute these expressions into the second given equation:
[tex]\[ 7x + 8y + 5z = 62 \][/tex]
Replacing [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[ 7(4k) + 8(3k) + 5(2k) = 62 \][/tex]
### Step 3: Simplify and Solve for [tex]\(k\)[/tex]
Let's simplify the equation step by step:
[tex]\[ 28k + 24k + 10k = 62 \][/tex]
[tex]\[ 62k = 62 \][/tex]
[tex]\[ k = 1 \][/tex]
### Step 4: Find the Values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]
Now, we substitute [tex]\(k = 1\)[/tex] back into the expressions for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[ x = 4k = 4 \][/tex]
[tex]\[ y = 3k = 3 \][/tex]
[tex]\[ z = 2k = 2 \][/tex]
So, the solution to the system of equations is:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = 3 \][/tex]
[tex]\[ z = 2 \][/tex]
Therefore, the answer to the problem is:
[tex]\[ \boxed{(4, 3, 2)} \][/tex]
These values satisfy the given system of equations.
