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Sagot :
Let's solve the problem step-by-step.
We are given that the sum of 2 times a number and 10 is the same as 6 times the difference of the number and 1. Let's denote the unknown number as [tex]\( x \)[/tex].
Mathematically, the given problem can be expressed as:
[tex]\[ 2x + 10 = 6(x - 1) \][/tex]
Now, let's solve this equation for [tex]\( x \)[/tex].
Step 1: Distribute the 6 on the right-hand side of the equation:
[tex]\[ 2x + 10 = 6x - 6 \][/tex]
Step 2: Get all the terms involving [tex]\( x \)[/tex] on one side of the equation and the constant terms on the other side. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 10 = 6x - 2x - 6 \][/tex]
Simplify:
[tex]\[ 10 = 4x - 6 \][/tex]
Step 3: Add 6 to both sides to get the constant term on the left-hand side:
[tex]\[ 10 + 6 = 4x \][/tex]
Simplify:
[tex]\[ 16 = 4x \][/tex]
Step 4: Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{16}{4} \][/tex]
Simplify:
[tex]\[ x = 4 \][/tex]
So, the number is 4. However, given the provided choices are -4, -1, and 2, we might want to reconsider if there may have been a potential misunderstanding in listing the answer choices.
To verify, let's check if our solution is correct:
Substitute [tex]\( x = 4 \)[/tex] back into the original condition:
[tex]\[ 2(4) + 10 = 6(4 - 1) \][/tex]
Calculate both sides:
[tex]\[ 8 + 10 = 6 \cdot 3 \][/tex]
[tex]\[ 18 = 18 \][/tex]
Since both sides are equal, [tex]\( x = 4 \)[/tex] is indeed the correct solution, but since 4 is not one of the provided answer choices, it suggests a mistake in the listed answer options.
If we are to select from the provided choices (-4, -1, 2), then checking each:
For [tex]\( x = -4 \)[/tex]:
[tex]\[ 2(-4) + 10 \neq 6(-4 - 1) \][/tex]
[tex]\[ -8 + 10 \neq 6 \cdot -5 \][/tex]
[tex]\[ 2 \neq -30 \][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[ 2(-1) + 10 \neq 6(-1 - 1) \][/tex]
[tex]\[ -2 + 10 \neq 6 \cdot -2 \][/tex]
[tex]\[ 8 \neq -12 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ 2(2) + 10 \neq 6(2 - 1) \][/tex]
[tex]\[ 4 + 10 \neq 6 \cdot 1 \][/tex]
[tex]\[ 14 \neq 6 \][/tex]
As observed, none of the provided choices (-4, -1, 2) solve the equation given!
Given that results from choices do not match, it would be occasion to raise a flag about inaccuracies in the answer choice listing (not a mathematical problem but preserving integrity of the choices).
Perhaps, next steps provide the best recognized mathematical solution explicitly: 4, aligning with provided checks.
We are given that the sum of 2 times a number and 10 is the same as 6 times the difference of the number and 1. Let's denote the unknown number as [tex]\( x \)[/tex].
Mathematically, the given problem can be expressed as:
[tex]\[ 2x + 10 = 6(x - 1) \][/tex]
Now, let's solve this equation for [tex]\( x \)[/tex].
Step 1: Distribute the 6 on the right-hand side of the equation:
[tex]\[ 2x + 10 = 6x - 6 \][/tex]
Step 2: Get all the terms involving [tex]\( x \)[/tex] on one side of the equation and the constant terms on the other side. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 10 = 6x - 2x - 6 \][/tex]
Simplify:
[tex]\[ 10 = 4x - 6 \][/tex]
Step 3: Add 6 to both sides to get the constant term on the left-hand side:
[tex]\[ 10 + 6 = 4x \][/tex]
Simplify:
[tex]\[ 16 = 4x \][/tex]
Step 4: Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{16}{4} \][/tex]
Simplify:
[tex]\[ x = 4 \][/tex]
So, the number is 4. However, given the provided choices are -4, -1, and 2, we might want to reconsider if there may have been a potential misunderstanding in listing the answer choices.
To verify, let's check if our solution is correct:
Substitute [tex]\( x = 4 \)[/tex] back into the original condition:
[tex]\[ 2(4) + 10 = 6(4 - 1) \][/tex]
Calculate both sides:
[tex]\[ 8 + 10 = 6 \cdot 3 \][/tex]
[tex]\[ 18 = 18 \][/tex]
Since both sides are equal, [tex]\( x = 4 \)[/tex] is indeed the correct solution, but since 4 is not one of the provided answer choices, it suggests a mistake in the listed answer options.
If we are to select from the provided choices (-4, -1, 2), then checking each:
For [tex]\( x = -4 \)[/tex]:
[tex]\[ 2(-4) + 10 \neq 6(-4 - 1) \][/tex]
[tex]\[ -8 + 10 \neq 6 \cdot -5 \][/tex]
[tex]\[ 2 \neq -30 \][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[ 2(-1) + 10 \neq 6(-1 - 1) \][/tex]
[tex]\[ -2 + 10 \neq 6 \cdot -2 \][/tex]
[tex]\[ 8 \neq -12 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ 2(2) + 10 \neq 6(2 - 1) \][/tex]
[tex]\[ 4 + 10 \neq 6 \cdot 1 \][/tex]
[tex]\[ 14 \neq 6 \][/tex]
As observed, none of the provided choices (-4, -1, 2) solve the equation given!
Given that results from choices do not match, it would be occasion to raise a flag about inaccuracies in the answer choice listing (not a mathematical problem but preserving integrity of the choices).
Perhaps, next steps provide the best recognized mathematical solution explicitly: 4, aligning with provided checks.
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