7.1 Solving equations with the variable on both sides Practice Solve each equation. Show all work. 1. \( 2 x+3=x+11 \) \( 3 \cdot 3 x-4=x+5 \) 5. \( 4 a+2=3 a+6 \) 6. \( -6 b-29=5 b-7 \) 7. \( -x+5=x-11 \) 8. \( 100-6 x=160-10 x \) \( 10.3-5 z=-9+z \)

Let's solve each equation step by step, showing all work. ### 1. Solve \( 2x + 3 = x + 11 \) **Step 1:** Subtract \( x \) from both sides: \[ 2x - x + 3 = 11 \] This simplifies to: \[ x + 3 = 11 \] **Step 2:** Subtract 3 from both sides: \[ x = 11 - 3 \] Thus, we find: \[ x = 8 \] ### 2. Solve \( 3 \cdot 3x - 4 = x + 5 \) **Step 1:** Simplify the left side: \[ 9x - 4 = x + 5 \] **Step 2:** Subtract \( x \) from both sides: \[ 9x - x - 4 = 5 \] This simplifies to: \[ 8x - 4 = 5 \] **Step 3:** Add 4 to both sides: \[ 8x = 5 + 4 \] Thus, we find: \[ 8x = 9 \] **Step 4:** Divide by 8: \[ x = \frac{9}{8} \] ### 3. Solve \( 4a + 2 = 3a + 6 \) **Step 1:** Subtract \( 3a \) from both sides: \[ 4a - 3a + 2 = 6 \] This simplifies to: \[ a + 2 = 6 \] **Step 2:** Subtract 2 from both sides: \[ a = 6 - 2 \] Thus, we find: \[ a = 4 \] ### 4. Solve \( -6b - 29 = 5b - 7 \) **Step 1:** Add \( 6b \) to both sides: \[ -29 = 5b + 6b - 7 \] This simplifies to: \[ -29 = 11b - 7 \] **Step 2:** Add 7 to both sides: \[ -29 + 7 = 11b \] Thus, we find: \[ -22 = 11b \] **Step 3:** Divide by 11: \[ b = -2 \] ### 5. Solve \( -x + 5 = x - 11 \) **Step 1:** Add \( x \) to both sides: \[ 5 = x + x - 11 \] This simplifies to: \[ 5 = 2x - 11 \] **Step 2:** Add 11 to both sides: \[ 5 + 11 = 2x \] Thus, we find: \[ 16 = 2x \] **Step 3:** Divide by 2: \[ x = 8 \] ### 6. Solve \( 100 - 6x = 160 - 10x \) **Step 1:** Add \( 10x \) to both sides: \[ 100 - 6x + 10x = 160 \] This simplifies to: \[ 100 + 4x = 160 \] **Step 2:** Subtract 100 from both sides: \[ 4x = 160 - 100 \] Thus, we find: \[ 4x = 60 \] **Step 3:** Divide by 4: \[ x = 15 \] ### 7. Solve \( 10.3 - 5z = -9 + z \) **Step 1:** Add \( 5z \) to both sides: \[ 10.3 = -9 + z + 5z \] This simplifies to: \[ 10.3 = -9 + 6z \] **Step 2:** Add 9 to both sides: \[ 10.3 + 9 = 6z \] Thus, we find: \[ 19.3 = 6z \] **Step 3:** Divide by 6: \[ z = \frac{19.3}{6} \approx 3.2167 \] ### Summary of Solutions 1. \( x = 8 \) 2. \( x = \frac{9}{8} \) 3. \( a = 4 \) 4. \( b = -2 \) 5. \( x = 8 \) 6. \( x = 15 \) 7. \( z \approx 3.2167 \)