Question: Find the intersection of the lines \( y = 3x + 1 \) and \( y = -2x + 11 \).
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- Solution: Apply the formula \( (a - b)^2 = a^2 - 2ab + b^2 \). Here, \( (2x)^2 - 2 \cdot 2x \cdot 3y + (3y)^2 = 4x^2 - 12xy + 9y^2 \). oxed{4x^2 - 12xy + 9y^2}
- Question: If \( x + rac{1}{x} = 5 \), what is the value of \( 2x^2 + rac{2}{x^2} \)?
- Solution: Square both sides: \( x^2 + 2 + rac{1}{x^2} = 25 \), so \( x^2 + rac{1}{x^2} = 23 \). Multiply by 2: \( 2x^2 + rac{2}{x^2} = 46 \). oxed{46}
- Solution: Set \( 3x + 1 = -2x + 11 \). Solve for \( x \): \( 5x = 10 \) â \( x = 2 \). Substitute into \( y = 3(2) + 1 = 7 \). The intersection is \( (2, 7) \). oxed{(2, 7)}
- Question: If \( m(m + n) = 24 \) and \( m + n = 8 \), what is the value of \( m \)?
- Solution: Substitute \( m + n = 8 \) into the first equation: \( m \cdot 8 = 24 \), so \( m = 3 \). oxed{3}Question: A sequence of six real numbers forms an arithmetic progression. The sum of the first three terms is 12, and the sum of the last three terms is 30. What is the sum of all six terms?
