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The distributive property is a key algebraic property that allows you to multiply a single term by each term within a parenthesis. In simple terms, if you have an equation like A(B + C), the distributive property lets you 'distribute' the A across by multiplying it with B and then with C, to get AB + AC.

For instance, in our exercise, you apply the distributive property to multiply 4 by both 1 and -3x, which gives us 4(1) - 4(3x). This step transforms the equation into a simpler form that is easier to solve because it breaks down expressions into simpler components. Picturing the distributive property as dispersing a numerical value across a wall of parentheses can often make the algebraic process more tangible for students.

Combining like terms is a technique used to simplify an algebraic expression or equation. Like terms are terms that have the exact same variable raised to the same power. Only the coefficients of these terms can differ. For example, 2x and 3x are like terms, but 2x and 2y are not.

In the problem, after applying the distributive property, you were left with the terms -12x and 7x on the same side. These are like terms because they both have the variable x raised to the first power. By combining them (adding or subtracting their coefficients), you simplify the equation to -5x + 4 = 9. This step makes it easier to isolate the variable in the next phase.

• Remember: combining like terms helps to streamline the equation, paving the way for easier problem-solving.

Isolating the variable is akin to finding the 'star of the show' in an equation. It's when you move all the terms except for the one containing the variable of interest to the other side of the equation. This is typically done using basic arithmetic operations: addition, subtraction, multiplication, and division, which are performed inversely.

Drawing from the worked example, after combining like terms, you're left with -5x + 4 = 9. To isolate x, subtract 4 from both sides of the equation, yielding -5x = 5. Isolating the variable sets the stage for solving the equation, which in this case, involves dividing both sides by -5 to find that x = -1.

• Isolation is a critical step; think of it as giving the variable its own 'solo performance' on the equation stage, allowing for a clear solution.