A student scored \( x \) points on his first test and \( y \) points on his second test. Since he did not do very well on the first one, he studied very hard for the second test. The increase in his score, \( y-x \), was 26 points. His average for the two tests, \( \frac{x+y}{2} \), was 68 points. What did he score on the first test? \( x= \)

To determine the score \( x \) on the first test, let's set up and solve the given equations step by step. 1. **Understanding the Problem:** - **First Test Score:** \( x \) - **Second Test Score:** \( y \) - **Increase in Score:** \( y - x = 26 \) points - **Average of Two Tests:** \( \frac{x + y}{2} = 68 \) points 2. **Setting Up the Equations:** - From the increase in score: \[ y = x + 26 \quad \text{(Equation 1)} \] - From the average score: \[ \frac{x + y}{2} = 68 \] Multiply both sides by 2 to eliminate the denominator: \[ x + y = 136 \quad \text{(Equation 2)} \] 3. **Substituting Equation 1 into Equation 2:** \[ x + (x + 26) = 136 \] Simplify: \[ 2x + 26 = 136 \] Subtract 26 from both sides: \[ 2x = 110 \] Divide by 2: \[ x = 55 \] 4. **Conclusion:** The student scored **55 points** on the first test. **Answer:** \( x = 55 \)