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Sagot :
To solve the system of equations:
[tex]\[ \begin{cases} x + 2y + 5z = 9 \\ x = y + 8 \\ z = x - 3 \end{cases} \][/tex]
we need to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations simultaneously. Let's proceed step by step:
### Step 1: Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]
From the second equation:
[tex]\[ x = y + 8 \][/tex]
### Step 2: Express [tex]\( z \)[/tex] in terms of [tex]\( x \)[/tex]
From the third equation:
[tex]\[ z = x - 3 \][/tex]
### Step 3: Substitute [tex]\( x \)[/tex] and [tex]\( z \)[/tex] into the first equation
Substitute [tex]\( x = y + 8 \)[/tex] and [tex]\( z = x - 3 \)[/tex] into the first equation:
[tex]\[ x + 2y + 5z = 9 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( y + 8 \)[/tex]:
[tex]\[ (y + 8) + 2y + 5z = 9 \][/tex]
Replace [tex]\( z \)[/tex] with [tex]\( x - 3 \)[/tex] which is [tex]\( (y + 8) - 3 \)[/tex]:
[tex]\[ (y + 8) + 2y + 5((y + 8) - 3) = 9 \][/tex]
Simplify [tex]\( z \)[/tex]:
[tex]\[ z = y + 5 \][/tex]
Now substitute [tex]\( z = y + 5 \)[/tex] back into the equation:
[tex]\[ (y + 8) + 2y + 5(y + 5) = 9 \][/tex]
### Step 4: Simplify and solve for [tex]\( y \)[/tex]
Combine like terms:
[tex]\[ y + 8 + 2y + 5y + 25 = 9 \][/tex]
Combine all the [tex]\( y \)[/tex] terms:
[tex]\[ 8y + 33 = 9 \][/tex]
Subtract 33 from both sides:
[tex]\[ 8y = -24 \][/tex]
Divide both sides by 8:
[tex]\[ y = -3 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex] and [tex]\( z \)[/tex]
Using [tex]\( y = -3 \)[/tex] in the equation [tex]\( x = y + 8 \)[/tex]:
[tex]\[ x = -3 + 8 \][/tex]
[tex]\[ x = 5 \][/tex]
Now using [tex]\( x = 5 \)[/tex] in the equation [tex]\( z = x - 3 \)[/tex]:
[tex]\[ z = 5 - 3 \][/tex]
[tex]\[ z = 2 \][/tex]
### Solution
The values of the variables that satisfy all the equations are:
[tex]\[ x = 5, \ y = -3, \ z = 2 \][/tex]
[tex]\[ \begin{cases} x + 2y + 5z = 9 \\ x = y + 8 \\ z = x - 3 \end{cases} \][/tex]
we need to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations simultaneously. Let's proceed step by step:
### Step 1: Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]
From the second equation:
[tex]\[ x = y + 8 \][/tex]
### Step 2: Express [tex]\( z \)[/tex] in terms of [tex]\( x \)[/tex]
From the third equation:
[tex]\[ z = x - 3 \][/tex]
### Step 3: Substitute [tex]\( x \)[/tex] and [tex]\( z \)[/tex] into the first equation
Substitute [tex]\( x = y + 8 \)[/tex] and [tex]\( z = x - 3 \)[/tex] into the first equation:
[tex]\[ x + 2y + 5z = 9 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( y + 8 \)[/tex]:
[tex]\[ (y + 8) + 2y + 5z = 9 \][/tex]
Replace [tex]\( z \)[/tex] with [tex]\( x - 3 \)[/tex] which is [tex]\( (y + 8) - 3 \)[/tex]:
[tex]\[ (y + 8) + 2y + 5((y + 8) - 3) = 9 \][/tex]
Simplify [tex]\( z \)[/tex]:
[tex]\[ z = y + 5 \][/tex]
Now substitute [tex]\( z = y + 5 \)[/tex] back into the equation:
[tex]\[ (y + 8) + 2y + 5(y + 5) = 9 \][/tex]
### Step 4: Simplify and solve for [tex]\( y \)[/tex]
Combine like terms:
[tex]\[ y + 8 + 2y + 5y + 25 = 9 \][/tex]
Combine all the [tex]\( y \)[/tex] terms:
[tex]\[ 8y + 33 = 9 \][/tex]
Subtract 33 from both sides:
[tex]\[ 8y = -24 \][/tex]
Divide both sides by 8:
[tex]\[ y = -3 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex] and [tex]\( z \)[/tex]
Using [tex]\( y = -3 \)[/tex] in the equation [tex]\( x = y + 8 \)[/tex]:
[tex]\[ x = -3 + 8 \][/tex]
[tex]\[ x = 5 \][/tex]
Now using [tex]\( x = 5 \)[/tex] in the equation [tex]\( z = x - 3 \)[/tex]:
[tex]\[ z = 5 - 3 \][/tex]
[tex]\[ z = 2 \][/tex]
### Solution
The values of the variables that satisfy all the equations are:
[tex]\[ x = 5, \ y = -3, \ z = 2 \][/tex]
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