At what angle lines 3𝑦 = 2𝑥 + 5 and 3𝑥 + 2𝑦 = 8 cut each other?

Sure, let's solve this step by step.

Step 1: Convert the equations into slope-intercept form (y = mx + b), where m is the slope of the line.

The first equation is already in this form: 3y = 2x + 5, which simplifies to y = (2/3)x + 5/3. So, the slope (m1) of the first line is 2/3.

The second equation can be rearranged to: 2y = -3x + 8, which simplifies to y = -(3/2)x + 4. So, the slope (m2) of the second line is -3/2.

Step 2: Use the formula for the angle θ between two lines: tan(θ) = |(m2 - m1) / (1 + m1*m2)|.

Substitute the slopes of the two lines into the formula: tan(θ) = |(-3/2 - 2/3) / (1 + 2/3*-3/2)|.

Simplify the expression to find the tangent of the angle.

Step 3: Use the inverse tangent function (also known as arctan or tan^-1) to find the angle θ.

Remember to convert the result from radians to degrees if necessary, since most calculators and programming languages use radians by default in their trigonometric functions.