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Sagot :
Let's solve the given system of equations:
[tex]\[ \begin{array}{l} 3x - 4y = 6 \quad \text{(Equation 1)} \\ x = \frac{4}{3} y + 2 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 1: Substitution
Since Equation 2 directly provides a value for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], we can substitute this expression into Equation 1.
From Equation 2:
[tex]\[ x = \frac{4}{3} y + 2 \][/tex]
Substitute this into Equation 1:
[tex]\[ 3\left(\frac{4}{3} y + 2\right) - 4y = 6 \][/tex]
### Step 2: Simplifying the Equation
First, distribute the 3:
[tex]\[ 3 \left(\frac{4}{3} y\right) + 3 \cdot 2 - 4y = 6 \][/tex]
[tex]\[ 4y + 6 - 4y = 6 \][/tex]
Now, simplify the equation:
[tex]\[ 4y - 4y + 6 = 6 \][/tex]
[tex]\[ 6 = 6 \][/tex]
### Step 3: Analyzing the Result
The result [tex]\( 6 = 6 \)[/tex] indicates that the equation is always true and does not depend on the value of [tex]\( y \)[/tex]. This means that there are infinitely many solutions, and the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be described as:
### Step 4: Expressing the Relationship
From Equation 2, we have:
[tex]\[ x = \frac{4}{3} y + 2 \][/tex]
This equation indicates a linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. For any value of [tex]\( y \)[/tex], we can compute the corresponding value of [tex]\( x \)[/tex] using this relationship.
### Final Solution
The system of equations can be represented by the equation:
[tex]\[ x = \frac{4}{3} y + 2 \][/tex]
Or, equivalently using decimal values:
[tex]\[ x = 1.33333333333333y + 2.0 \][/tex]
This expression provides the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in this system, representing an infinite number of solutions along the line described by this equation.
[tex]\[ \begin{array}{l} 3x - 4y = 6 \quad \text{(Equation 1)} \\ x = \frac{4}{3} y + 2 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 1: Substitution
Since Equation 2 directly provides a value for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], we can substitute this expression into Equation 1.
From Equation 2:
[tex]\[ x = \frac{4}{3} y + 2 \][/tex]
Substitute this into Equation 1:
[tex]\[ 3\left(\frac{4}{3} y + 2\right) - 4y = 6 \][/tex]
### Step 2: Simplifying the Equation
First, distribute the 3:
[tex]\[ 3 \left(\frac{4}{3} y\right) + 3 \cdot 2 - 4y = 6 \][/tex]
[tex]\[ 4y + 6 - 4y = 6 \][/tex]
Now, simplify the equation:
[tex]\[ 4y - 4y + 6 = 6 \][/tex]
[tex]\[ 6 = 6 \][/tex]
### Step 3: Analyzing the Result
The result [tex]\( 6 = 6 \)[/tex] indicates that the equation is always true and does not depend on the value of [tex]\( y \)[/tex]. This means that there are infinitely many solutions, and the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be described as:
### Step 4: Expressing the Relationship
From Equation 2, we have:
[tex]\[ x = \frac{4}{3} y + 2 \][/tex]
This equation indicates a linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. For any value of [tex]\( y \)[/tex], we can compute the corresponding value of [tex]\( x \)[/tex] using this relationship.
### Final Solution
The system of equations can be represented by the equation:
[tex]\[ x = \frac{4}{3} y + 2 \][/tex]
Or, equivalently using decimal values:
[tex]\[ x = 1.33333333333333y + 2.0 \][/tex]
This expression provides the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in this system, representing an infinite number of solutions along the line described by this equation.
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