\#7 The perimeter of a triangle is 73 inches. If the second side is 5 inches longer than twice the first side, and the third side is 4 inches less than three times the first side, how long is each side?

To solve this problem, we can use the information given about the sides of the triangle to set up a system of equations. Let's denote the length of the first side as \( x \) inches. Given: - The second side is 5 inches longer than twice the first side: \( 2x + 5 \) inches. - The third side is 4 inches less than three times the first side: \( 3x - 4 \) inches. The perimeter of the triangle is 73 inches, so we can set up the equation: \[ x + (2x + 5) + (3x - 4) = 73 \] Simplifying the equation: \[ 6x + 1 = 73 \] Now, we can solve for \( x \) to find the length of the first side. Solve the equation by following steps: - step0: Solve for \(x\): \(6x+1=73\) - step1: Move the constant to the right side: \(6x=73-1\) - step2: Subtract the numbers: \(6x=72\) - step3: Divide both sides: \(\frac{6x}{6}=\frac{72}{6}\) - step4: Divide the numbers: \(x=12\) The length of the first side is 12 inches. Now, we can find the lengths of the second and third sides using the given information: - Second side: \( 2x + 5 = 2(12) + 5 = 24 + 5 = 29 \) inches. - Third side: \( 3x - 4 = 3(12) - 4 = 36 - 4 = 32 \) inches. Therefore, the lengths of the sides of the triangle are: - First side: 12 inches. - Second side: 29 inches. - Third side: 32 inches.