91Ó°ÊÓ

Isolating the variable is an essential step in solving equations, particularly for linear equations like those provided in the exercise. To isolate a variable, it means to "get the variable by itself" on one side of the equation. This usually involves performing a series of algebraic operations to move other terms to the opposite side of the equation. For example, in equation (a) from the original exercise: - We start with: \(3x + 4y = 5.2\). - To isolate \(y\), subtract \(3x\) from both sides, which gives \(4y = 5.2 - 3x\). - Lastly, divide each term by 4, resulting in the expression \(y = \frac{5.2 - 3x}{4}\).By isolating \(y\), you can clearly see how changes in the value of \(x\) affect \(y\), allowing for a better understanding of the relationship between these variables.

Expressing a variable in terms of another involves rearranging an equation so that one variable is isolated on one side, often written as something like \(y = f(x)\). In this process, you'll transform an equation to explicitly show how one variable depends on others. This process is vital for understanding the functional relationship between variables, making it easier to graph equations or substitute values.In the exercise, both equations are eventually expressed with \(y\) as a function of \(x\): - For equation (a), we achieve: \(y = 1.3 - 0.75x\).- For equation (b), we achieve: \(y = \frac{2x}{3} + 5\).Understanding how to express variables effectively not only aids in solving equations but also helps in interpreting and predicting the behavior of linear relationships.

The distributive property is a fundamental algebraic property used to simplify expressions and equations. It states that multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then summing the products. Mathematically, it's expressed as \(a(b + c) = ab + ac\).Applying this property can help simplify equations, making it easier to solve for a particular variable. For instance, in equation (b): - The initial step is \(3(y - 5) = 2x\).- Using the distributive property, distribute \(3\) across \(y - 5\): this transforms the equation into \(3y - 15 = 2x\).After distributing, the equation becomes simpler to work with, allowing you to proceed with isolating the variable you wish to solve for. Mastering the distributive property is crucial for tackling more complex algebraic expressions and equations efficiently.