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Sagot :
To solve the given proportion and derive the correct linear equation, let's go through a step-by-step process.
We are given the proportion:
[tex]\[ \frac{x - 9}{x - 3} = \frac{2}{5} \][/tex]
To remove the fraction, we can cross-multiply. Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other.
So, we multiply:
[tex]\[ 5 \times (x - 9) = 2 \times (x - 3) \][/tex]
This gives us:
[tex]\[ 5(x - 9) = 2(x - 3) \][/tex]
Now, we distribute the numbers through the parentheses on both sides:
[tex]\[ 5x - 45 = 2x - 6 \][/tex]
Our goal is to isolate [tex]\(x\)[/tex] on one side of the equation. To do that, we'll move all the terms involving [tex]\(x\)[/tex] to one side and the constant terms to the other side.
First, let's subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ 5x - 2x - 45 = -6 \][/tex]
[tex]\[ 3x - 45 = -6 \][/tex]
Next, let's add 45 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ 3x - 45 + 45 = -6 + 45 \][/tex]
[tex]\[ 3x = 39 \][/tex]
Finally, we solve for [tex]\(x\)[/tex] by dividing both sides by 3:
[tex]\[ x = \frac{39}{3} \][/tex]
[tex]\[ x = 13 \][/tex]
So, the derived equation before simplifying further was:
[tex]\[ 5x - 45 = 2x - 6 \][/tex]
Thus, the correct linear equation matching this derived form is:
[tex]\[ 2x - 6 = 5x - 45 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{A} \: 2x - 6 = 5x - 45 \][/tex]
We are given the proportion:
[tex]\[ \frac{x - 9}{x - 3} = \frac{2}{5} \][/tex]
To remove the fraction, we can cross-multiply. Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other.
So, we multiply:
[tex]\[ 5 \times (x - 9) = 2 \times (x - 3) \][/tex]
This gives us:
[tex]\[ 5(x - 9) = 2(x - 3) \][/tex]
Now, we distribute the numbers through the parentheses on both sides:
[tex]\[ 5x - 45 = 2x - 6 \][/tex]
Our goal is to isolate [tex]\(x\)[/tex] on one side of the equation. To do that, we'll move all the terms involving [tex]\(x\)[/tex] to one side and the constant terms to the other side.
First, let's subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ 5x - 2x - 45 = -6 \][/tex]
[tex]\[ 3x - 45 = -6 \][/tex]
Next, let's add 45 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ 3x - 45 + 45 = -6 + 45 \][/tex]
[tex]\[ 3x = 39 \][/tex]
Finally, we solve for [tex]\(x\)[/tex] by dividing both sides by 3:
[tex]\[ x = \frac{39}{3} \][/tex]
[tex]\[ x = 13 \][/tex]
So, the derived equation before simplifying further was:
[tex]\[ 5x - 45 = 2x - 6 \][/tex]
Thus, the correct linear equation matching this derived form is:
[tex]\[ 2x - 6 = 5x - 45 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{A} \: 2x - 6 = 5x - 45 \][/tex]
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