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Sagot :
Absolutely! Let's solve the equation step-by-step:
Given equation:
[tex]\[ -\frac{2x}{2x - 10} + 3 = \frac{10}{2x - 10} \][/tex]
First, we want to simplify and solve this rational equation. To make it simpler, let's rewrite it with a common denominator. Notice that both fractions share the denominator [tex]\(2x - 10\)[/tex].
Rewrite the left-hand side:
[tex]\[ 3 = \frac{3(2x - 10)}{2x - 10} \][/tex]
So the equation becomes:
[tex]\[ -\frac{2x}{2x - 10} + \frac{3(2x - 10)}{2x - 10} = \frac{10}{2x - 10} \][/tex]
Combine the fractions on the left-hand side:
[tex]\[ \frac{-2x + 3(2x - 10)}{2x - 10} = \frac{10}{2x - 10} \][/tex]
Simplify the numerator on the left-hand side:
[tex]\[ -2x + 3(2x - 10) = -2x + 6x - 30 = 4x - 30 \][/tex]
Thus, the equation becomes:
[tex]\[ \frac{4x - 30}{2x - 10} = \frac{10}{2x - 10} \][/tex]
Since the denominators are equal, we can set the numerators equal to each other:
[tex]\[ 4x - 30 = 10 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ 4x - 30 = 10 \\ 4x = 40 \\ x = 10 \][/tex]
Before we conclude that [tex]\(x = 10\)[/tex] is indeed a solution, we must check if it makes any denominators zero, because division by zero is undefined.
Check the denominator [tex]\(2x - 10\)[/tex]:
If [tex]\(x = 10\)[/tex],
[tex]\[ 2(10) - 10 = 20 - 10 = 10 \neq 0 \][/tex]
Since the denominator does not become zero, [tex]\(x = 10\)[/tex] is a valid solution.
Therefore, the solution to the equation is:
[tex]\[ x = 10 \][/tex]
Given equation:
[tex]\[ -\frac{2x}{2x - 10} + 3 = \frac{10}{2x - 10} \][/tex]
First, we want to simplify and solve this rational equation. To make it simpler, let's rewrite it with a common denominator. Notice that both fractions share the denominator [tex]\(2x - 10\)[/tex].
Rewrite the left-hand side:
[tex]\[ 3 = \frac{3(2x - 10)}{2x - 10} \][/tex]
So the equation becomes:
[tex]\[ -\frac{2x}{2x - 10} + \frac{3(2x - 10)}{2x - 10} = \frac{10}{2x - 10} \][/tex]
Combine the fractions on the left-hand side:
[tex]\[ \frac{-2x + 3(2x - 10)}{2x - 10} = \frac{10}{2x - 10} \][/tex]
Simplify the numerator on the left-hand side:
[tex]\[ -2x + 3(2x - 10) = -2x + 6x - 30 = 4x - 30 \][/tex]
Thus, the equation becomes:
[tex]\[ \frac{4x - 30}{2x - 10} = \frac{10}{2x - 10} \][/tex]
Since the denominators are equal, we can set the numerators equal to each other:
[tex]\[ 4x - 30 = 10 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ 4x - 30 = 10 \\ 4x = 40 \\ x = 10 \][/tex]
Before we conclude that [tex]\(x = 10\)[/tex] is indeed a solution, we must check if it makes any denominators zero, because division by zero is undefined.
Check the denominator [tex]\(2x - 10\)[/tex]:
If [tex]\(x = 10\)[/tex],
[tex]\[ 2(10) - 10 = 20 - 10 = 10 \neq 0 \][/tex]
Since the denominator does not become zero, [tex]\(x = 10\)[/tex] is a valid solution.
Therefore, the solution to the equation is:
[tex]\[ x = 10 \][/tex]
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