Final answer:
The multiplication property of equality and simplification by cancelling common factors are used to solve equations like 4y = 6 and manipulating proportions involving scale factors. To find actual dimensions when a scale factor is given, set up a proportion and cross-multiply to solve for the missing value.
Step-by-step explanation:
The property used to solve the equations 4y = 6 and 1/4 x 4y = 6 x 1/4 is the multiplication property of equality, which states that if you multiply both sides of an equation by the same number, the two sides remain equal. Additionally, when solving proportions or equations involving fractions, often we use the property of multiplication that allows us to simplify fractions by cancelling out common factors in the numerator and the denominator.
For instance, if we want to simplify 1/4 x 4y, we notice that 4 in the numerator and 4 in the denominator cancel each other out, leaving us with y. Similarly, when we multiply 6 x 1/4, we simply get 6/4 which simplifies to 3/2 or 1.5. This gives us the simple equation y = 3/2.
To solve problems involving scale factors and proportions, you can set up a proportion like 1:2=4:x. Here, you would cross-multiply to find the value of x, which represents the actual dimension in relation to the given scale dimension. For an equation like 1/48=w/16, you would also cross-multiply to find the value of w, which again represents the actual dimension when a scale factor is involved.