Pregunta
An equation is shown below. \[ 3(x-2)+7 x=\frac{1}{2}(6 x-2) \] How many solutions, if any. does the equation have?
Ask by Wright Evans. in the United States
Jan 21,2025
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Respuesta verificada por el tutor
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The equation has one solution: \( x = \frac{5}{7} \) or approximately \( 0.714 \).
Solución
Solve the equation \( 3(x-2)+7x=\frac{1}{2}(6x-2) \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3\left(x-2\right)+7x=\frac{1}{2}\left(6x-2\right)\)
- step1: Multiply the terms:
\(3\left(x-2\right)+7x=3x-1\)
- step2: Move the expression to the left side:
\(3\left(x-2\right)+7x-\left(3x-1\right)=0\)
- step3: Calculate:
\(-5+7x=0\)
- step4: Move the constant to the right side:
\(7x=0+5\)
- step5: Remove 0:
\(7x=5\)
- step6: Divide both sides:
\(\frac{7x}{7}=\frac{5}{7}\)
- step7: Divide the numbers:
\(x=\frac{5}{7}\)
The equation \(3(x-2)+7x=\frac{1}{2}(6x-2)\) has one solution, which is \(x=\frac{5}{7}\) or approximately \(0.714285\).
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución The Deep Dive
To find the number of solutions for the equation \( 3(x-2)+7 x=\frac{1}{2}(6 x-2) \), let's start by simplifying it step by step. 1. Distribute on the left side: \[ 3x - 6 + 7x = \frac{1}{2}(6x - 2) \] Combine like terms: \[ 10x - 6 = \frac{1}{2}(6x - 2) \] 2. Now simplify the right side: \[ \frac{1}{2}(6x - 2) = 3x - 1 \] 3. Now the equation is: \[ 10x - 6 = 3x - 1 \] 4. Move all terms involving \( x \) to one side and constant terms to the other side: \[ 10x - 3x = -1 + 6 \] \[ 7x = 5 \] 5. Solve for \( x \): \[ x = \frac{5}{7} \] Since we found a unique solution \( x = \frac{5}{7} \), the equation has exactly one solution!
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