\( \begin{aligned} 6 x-10+3 & =12 x \\ 6 x-7 & =12 x \\ \text { e. } \frac{4 x+5}{2 x+1} & =\frac{16}{5}\end{aligned} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(6x-7=12x\) - step1: Move the variable to the left side: \(6x-7-12x=0\) - step2: Subtract the terms: \(-6x-7=0\) - step3: Move the constant to the right side: \(-6x=0+7\) - step4: Remove 0: \(-6x=7\) - step5: Change the signs: \(6x=-7\) - step6: Divide both sides: \(\frac{6x}{6}=\frac{-7}{6}\) - step7: Divide the numbers: \(x=-\frac{7}{6}\) Solve the equation \( 6x-10+3=12x \). Solve the equation by following steps: - step0: Solve for \(x\): \(6x-10+3=12x\) - step1: Add the numbers: \(6x-7=12x\) - step2: Move the variable to the left side: \(6x-7-12x=0\) - step3: Subtract the terms: \(-6x-7=0\) - step4: Move the constant to the right side: \(-6x=0+7\) - step5: Remove 0: \(-6x=7\) - step6: Change the signs: \(6x=-7\) - step7: Divide both sides: \(\frac{6x}{6}=\frac{-7}{6}\) - step8: Divide the numbers: \(x=-\frac{7}{6}\) Solve the equation \( \frac{4x+5}{2x+1}=\frac{16}{5} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{4x+5}{2x+1}=\frac{16}{5}\) - step1: Find the domain: \(\frac{4x+5}{2x+1}=\frac{16}{5},x\neq -\frac{1}{2}\) - step2: Cross multiply: \(\left(4x+5\right)\times 5=\left(2x+1\right)\times 16\) - step3: Simplify the equation: \(5\left(4x+5\right)=16\left(2x+1\right)\) - step4: Calculate: \(20x+25=16\left(2x+1\right)\) - step5: Calculate: \(20x+25=32x+16\) - step6: Move the expression to the left side: \(20x+25-\left(32x+16\right)=0\) - step7: Calculate: \(-12x+9=0\) - step8: Move the constant to the right side: \(-12x=0-9\) - step9: Remove 0: \(-12x=-9\) - step10: Change the signs: \(12x=9\) - step11: Divide both sides: \(\frac{12x}{12}=\frac{9}{12}\) - step12: Divide the numbers: \(x=\frac{3}{4}\) - step13: Check if the solution is in the defined range: \(x=\frac{3}{4},x\neq -\frac{1}{2}\) - step14: Find the intersection: \(x=\frac{3}{4}\) Let's solve the given equations step by step. 1. **First Equation:** \[ 6x - 10 + 3 = 12x \] Simplifying this: \[ 6x - 7 = 12x \] Rearranging gives: \[ -7 = 12x - 6x \implies -7 = 6x \implies x = -\frac{7}{6} \] Thus, the solution for the first equation is: \[ x = -\frac{7}{6} \quad \text{or approximately } -1.1667 \] 2. **Second Equation:** \[ 6x - 7 = 12x \] Rearranging gives: \[ -7 = 12x - 6x \implies -7 = 6x \implies x = -\frac{7}{6} \] The solution for the second equation is the same: \[ x = -\frac{7}{6} \quad \text{or approximately } -1.1667 \] 3. **Third Equation:** \[ \frac{4x + 5}{2x + 1} = \frac{16}{5} \] Cross-multiplying gives: \[ 5(4x + 5) = 16(2x + 1) \] Expanding both sides: \[ 20x + 25 = 32x + 16 \] Rearranging gives: \[ 25 - 16 = 32x - 20x \implies 9 = 12x \implies x = \frac{3}{4} \] Thus, the solution for the third equation is: \[ x = \frac{3}{4} \quad \text{or } 0.75 \] ### Summary of Solutions: - For the first equation: \( x = -\frac{7}{6} \) - For the second equation: \( x = -\frac{7}{6} \) - For the third equation: \( x = \frac{3}{4} \)