This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Understand the relationship between the area of an object, the area of its image, and the determinant of the transformation matrix. The area of the image (A_image) is related to the area of the object (A_object) by the absolute value of the determinant of the transformation matrix (M). A_image = |(M)| × A_object Step 2: Identify the given values. Area of object (A_object) = 9 cm^2 Area of image (A_image) = 54 cm^2 Transformation matrix (M) = 2 & x \\ 2 & 1 Step 3: Calculate the determinant of the transformation matrix. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (M) = (2)(1) - (x)(2) = 2 - 2x Step 4: Substitute the values into the area relationship formula. 54 = |2 - 2x| × 9 Step 5: Solve for |2 - 2x|. Divide both sides by 9: (54)/(9) = |2 - 2x| 6 = |2 - 2x| Step 6: Solve for x. The equation |2 - 2x| = 6 implies two possibilities: Case 1: 2 - 2x = 6 -2x = 6 - 2 -2x = 4 x = (4)/(-2) x = -2 Case 2: 2 - 2x = -6 -2x = -6 - 2 -2x = -8 x = (-8)/(-2) x = 4 Both values of x satisfy the given conditions. The value of x can be -2 or 4. That's 2 down. 3 left today — send the next one.