Let's solve each of the equations one by one.
1. Solution to the linear equation \(4b + 6 = 2 - b + 4\)
First, simplify the right side: \[ 2 - b + 4 = 6 - b \] Now, the equation is: \[ 4b + 6 = 6 - b \] Add \(b\) to both sides: \[ 4b + b + 6 = 6 \] This simplifies to: \[ 5b + 6 = 6 \] Subtract 6 from both sides: \[ 5b = 0 \] Dividing by 5: \[ b = 0 \]
Answer: \(b = 0\)
2. Number to eliminate fractions: \(6 - x + \frac{3}{4}x + \frac{1}{3} = \frac{1}{2}x + 5\)
The denominators are 4, 3, and 2. The least common multiple (LCM) of these denominators is 12.
Answer: 12
3. Number to eliminate fractions: \(m - \frac{3}{4}m - \frac{1}{2} = 2 \cdot \frac{1}{4}m + 2 + m\)
The denominators are 4 and 2. The least common multiple (LCM) of these denominators is 4.
Answer: 4
4. Solution to the linear equation \(d - 10 - 2d + 7 = 8 + d - 10 - 3d\)
Simplify both sides: \[ -d - 3 = -2d - 2 \] Adding \(2d\) to both sides: \[ d - 3 = -2 \] Adding 3 to both sides: \[ d = 1 \]
Answer: \(d = 1\)
5. Solution to the linear equation \(x - \frac{2}{3}x - \frac{1}{2} = \frac{1}{3} + \frac{5}{6}x\)
Combine like terms: \[ \left(1 - \frac{2}{3}\right)x - \frac{1}{2} = \frac{1}{3} + \frac{5}{6}x \] This simplifies to: \[ \frac{1}{3}x - \frac{1}{2} = \frac{1}{3} + \frac{5}{6}x \] After solving: Multiply through by 6 to eliminate fractions. \[ 2x - 3 = 2 + 5x \] Rearranging gives: \[ -3 - 2 = 5x - 2x \] \[ -5 = 3x \] \[ x = -\frac{5}{3} \] This doesn’t match any of the answers provided. However, let's verify the provided options. I apologize, there seems to be a potential mistake with my calculations. Please accept my answer reflecting that the solution seems ambiguous.
Answer: (Determine from options) Not clear from provided options.
6. Use properties to rewrite the equation correctly.
The equations with the same solutions include:
- \(x + \frac{8}{5}x + \frac{2}{3} = \frac{1}{2} - \frac{1}{5}x\).
- \(18x + 20 + x = 15 - 6x\).
- \(12x + 30x = -5\).
From your list, I will choose:
- \(x + \frac{2}{3} + x\) both sides can be interpreted.
- The expressions manipulated from \(x + \frac{8}{5}x + \frac{2}{3}\) are valid.
- The restructuring of combining additional equations like resulted.
Answer: Choose appropriate from options specifications made.
7. Solve the equation \( -3x + 1 + 10x = x + 4\)
Combine terms on the left: \[ 7x + 1 = x + 4 \] Now, subtract \(x\) from both sides: \[ 6x + 1 = 4 \] Subtracting 1: \[ 6x = 3 \] So: \[ x = \frac{1}{2} = x equals \frac{1}{2} \]
Answer: \(x = \frac{1}{2}\)
8. Solve the equation \(y + 6 = -3y + 26\)
Adding \(3y\) to both sides gives: \[ y + 3y + 6 = 26 \] This simplifies to: \[ 4y + 6 = 26 \] Subtract 6: \[ 4y = 20 \] And dividing both sides by 4: \[ y = 5 \]
Answer: \(y = 5\)
9. Justification for the step from line 2 to line 3.
The step simplified the equation: From -4x to -6x, this follows by performing isolated terms on either sides, indicative:
Answer: combining like terms on one side of the equation
10. Solution to the equation \(2.8y + 6 + 0.2y = 5y - 14\)
Combine like terms on the left: \[ 3y + 6 = 5y - 14 \] Subtract 3y from both sides: \[ 6 = 2y - 14 \] Adding 14: \[ 20 = 2y \] Dividing by 2: \[ y = 10 \]
Answer: \(y = 10\)
Overall, please review the above responses, specifying options mentioned if discrepancy exists from these findings based on direct inquiries which relate to resolution taking placed.
2 answers