Question

Solve the system by the method of substitution. $$\left\\{\begin{aligned} \frac{1}{5} x+\frac{1}{2} y &=8 \\ x+y &=20 \end{aligned}\right.$$

Short answer

The solution to the system of equations is \(x = \frac{20}{3}\) and \(y = \frac{40}{3}\).

Step-by-step solution

Expressing one variable in term of others

From the second provided equation: \(x + y = 20 \), solve for one variable: \(x = 20 - y\).

Substituting the obtained equation into the other

Substitute \(x = 20 - y\) into the first equation: \(\frac{1}{5}*(20 - y) + \frac{1}{2} * y = 8\). Simplify it to find the numerical value of y.

Solving for y

On simplifying: \(4 - \frac{1}{5} * y + \frac{1}{2} * y = 8 \) we get, \(\frac{3}{10} * y = 4 => y = \frac{40}{3}\).

Substituting y into the another equation

Substitute the value of y:=\frac{40}{3} into the second equation: \(x + \frac{40}{3} = 20\). Solve it to find the value of x.

Solving for x

On simplifying: \(x = 20 - \frac{40}{3} = \frac{20}{3}\).

Key concepts

Substitution Method

The substitution method is a powerful tool in algebra, especially when solving systems of linear equations. This method involves expressing one variable in terms of another and then substituting this expression into another equation within the system. This gives us a single equation with one unknown, which is usually simpler to solve.

For instance, given a system of equations such as:
\begin{align*}\frac{1}{5} x + \frac{1}{2} y &= 8 \x + y &= 20\end{align*},the substitution method starts by isolating one variable, like getting an expression for \( x \) from the second equation, \( x = 20 - y \).This expression for \( x \) is then substituted into the first equation, replacing the \( x \) there, which simplifies the problem to one equation with only \( y \) as an unknown. Once \( y \) is found, it is substituted back into the expression for \( x \) to find its value. The substitution method is ideal for systems where one equation can be easily solved for one variable. It also helps when avoiding the potential messiness of other methods, such as the elimination method, which can involve manipulating both equations extensively.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables, and arithmetic operations combined to represent a quantity. Expressions differ from equations in that they don't have an equality sign and hence do not state that two things are the same. Instead, they're like sentences that describe certain values or relationships.

An example of an algebraic expression is \( \frac{1}{5}x + \frac{1}{2}y \), which indicates a sum of two terms involving variables \( x \) and \( y \). These expressions become especially important in the context of the substitution method, as we create them when expressing one variable in terms of another, such as \( x = 20 - y \).It's crucial to handle algebraic expressions with care, as simplifying or modifying them is a critical step in solving equations. The ability to manipulate these expressions is foundational to success in algebra.

Simplifying Equations

Simplifying equations is an essential skill in algebra that makes solving them more manageable. It involves reducing equations to their simplest form by combining like terms, eliminating fractions, and simplifying expressions. It's like cleaning up a messy room so you can easily find what you're looking for.

When we face an equation after substitution, like \( \frac{1}{5}(20 - y) + \frac{1}{2}y = 8 \), simplifying helps to reveal the core relationship between variables. By multiplying out parentheses, finding common denominators, and combining terms, we transform this into \( 4 - \frac{1}{5}y + \frac{1}{2}y = 8 \), and eventually into \( \frac{3}{10}y = 4 \). Once the equation is simplified, finding the value of \( y \) becomes a less daunting task.Simplification is about making the problem as easy as possible to solve, which not only makes the current work less error-prone but also aids in understanding the underlying structure of the mathematical relationship.