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Sagot :
To solve the system of equations:
[tex]\[ \begin{align*} 1) &\quad 5 - 3y = 18, \\ 2) &\quad 3s^2 - y = 12, \end{align*} \][/tex]
we will first find the value of [tex]\(y\)[/tex] from the first equation, and then use it to find [tex]\(s\)[/tex] from the second equation. Let's proceed step-by-step.
### Step 1: Solve for [tex]\(y\)[/tex] from the first equation
The first equation is:
[tex]\[ 5 - 3y = 18 \][/tex]
Subtract 5 from both sides to isolate the term involving [tex]\(y\)[/tex]:
[tex]\[ -3y = 18 - 5 \][/tex]
Simplify the right-hand side:
[tex]\[ -3y = 13 \][/tex]
Divide both sides by -3 to solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{13}{-3} \][/tex]
So,
[tex]\[ y = -4.\overline{3} \][/tex]
### Step 2: Substitute [tex]\(y\)[/tex] into the second equation to solve for [tex]\(s\)[/tex]
The second equation is:
[tex]\[ 3s^2 - y = 12 \][/tex]
Substitute [tex]\(y = -4.\overline{3}\)[/tex] into the equation:
[tex]\[ 3s^2 - \left(-4.\overline{3}\right) = 12 \][/tex]
Simplify the left-hand side:
[tex]\[ 3s^2 + 4.\overline{3} = 12 \][/tex]
Next, we need to isolate [tex]\(s^2\)[/tex]. Subtract [tex]\(4.\overline{3}\)[/tex] from both sides:
[tex]\[ 3s^2 = 12 - 4.\overline{3} \][/tex]
We know that [tex]\(4.\overline{3}\)[/tex] is equivalent to [tex]\(\frac{13}{3}\)[/tex], so the equation becomes:
[tex]\[ 3s^2 = 12 - \frac{13}{3} \][/tex]
Convert 12 into a fraction with the same denominator:
[tex]\[ 12 = \frac{36}{3} \][/tex]
Now, subtract the fractions:
[tex]\[ 3s^2 = \frac{36}{3} - \frac{13}{3} \][/tex]
Combine the fractions:
[tex]\[ 3s^2 = \frac{36 - 13}{3} \][/tex]
Simplify the numerator:
[tex]\[ 3s^2 = \frac{23}{3} \][/tex]
Divide both sides by 3 to solve for [tex]\(s^2\)[/tex]:
[tex]\[ s^2 = \frac{23}{9} \][/tex]
### Step 3: Solve for [tex]\(s\)[/tex]
To find [tex]\(s\)[/tex], take the square root of both sides:
[tex]\[ s = \sqrt{\frac{23}{9}} \][/tex]
Simplify by finding the square root of the numerator and the denominator separately:
[tex]\[ s = \frac{\sqrt{23}}{\sqrt{9}} = \frac{\sqrt{23}}{3} \][/tex]
Thus, the numerical value of [tex]\(s\)[/tex] is approximately:
[tex]\[ s \approx 1.5986 \][/tex]
### Summary of Solutions
Therefore, the solutions are:
[tex]\[ y \approx -4.3333, \quad s^2 \approx 2.5556, \quad s \approx 1.5986 \][/tex]
These steps provide a clear and detailed solution to the given system of equations.
[tex]\[ \begin{align*} 1) &\quad 5 - 3y = 18, \\ 2) &\quad 3s^2 - y = 12, \end{align*} \][/tex]
we will first find the value of [tex]\(y\)[/tex] from the first equation, and then use it to find [tex]\(s\)[/tex] from the second equation. Let's proceed step-by-step.
### Step 1: Solve for [tex]\(y\)[/tex] from the first equation
The first equation is:
[tex]\[ 5 - 3y = 18 \][/tex]
Subtract 5 from both sides to isolate the term involving [tex]\(y\)[/tex]:
[tex]\[ -3y = 18 - 5 \][/tex]
Simplify the right-hand side:
[tex]\[ -3y = 13 \][/tex]
Divide both sides by -3 to solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{13}{-3} \][/tex]
So,
[tex]\[ y = -4.\overline{3} \][/tex]
### Step 2: Substitute [tex]\(y\)[/tex] into the second equation to solve for [tex]\(s\)[/tex]
The second equation is:
[tex]\[ 3s^2 - y = 12 \][/tex]
Substitute [tex]\(y = -4.\overline{3}\)[/tex] into the equation:
[tex]\[ 3s^2 - \left(-4.\overline{3}\right) = 12 \][/tex]
Simplify the left-hand side:
[tex]\[ 3s^2 + 4.\overline{3} = 12 \][/tex]
Next, we need to isolate [tex]\(s^2\)[/tex]. Subtract [tex]\(4.\overline{3}\)[/tex] from both sides:
[tex]\[ 3s^2 = 12 - 4.\overline{3} \][/tex]
We know that [tex]\(4.\overline{3}\)[/tex] is equivalent to [tex]\(\frac{13}{3}\)[/tex], so the equation becomes:
[tex]\[ 3s^2 = 12 - \frac{13}{3} \][/tex]
Convert 12 into a fraction with the same denominator:
[tex]\[ 12 = \frac{36}{3} \][/tex]
Now, subtract the fractions:
[tex]\[ 3s^2 = \frac{36}{3} - \frac{13}{3} \][/tex]
Combine the fractions:
[tex]\[ 3s^2 = \frac{36 - 13}{3} \][/tex]
Simplify the numerator:
[tex]\[ 3s^2 = \frac{23}{3} \][/tex]
Divide both sides by 3 to solve for [tex]\(s^2\)[/tex]:
[tex]\[ s^2 = \frac{23}{9} \][/tex]
### Step 3: Solve for [tex]\(s\)[/tex]
To find [tex]\(s\)[/tex], take the square root of both sides:
[tex]\[ s = \sqrt{\frac{23}{9}} \][/tex]
Simplify by finding the square root of the numerator and the denominator separately:
[tex]\[ s = \frac{\sqrt{23}}{\sqrt{9}} = \frac{\sqrt{23}}{3} \][/tex]
Thus, the numerical value of [tex]\(s\)[/tex] is approximately:
[tex]\[ s \approx 1.5986 \][/tex]
### Summary of Solutions
Therefore, the solutions are:
[tex]\[ y \approx -4.3333, \quad s^2 \approx 2.5556, \quad s \approx 1.5986 \][/tex]
These steps provide a clear and detailed solution to the given system of equations.
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