Problem 99 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} y=3 x \\ 6 x-2 y=0 \end{array}\right. $$

Step-by-Step Solution

Answer

Any \(x\) and corresponding \(y = 3x\).

Step 1: Substitute the Expression

The first equation is already solved for y: \(y = 3x\). Substitute \(3x\) for \(y\) in the second equation.

Step 2: Substitute into the Second Equation

Replace \(y\) in the second equation \(6x - 2y = 0\) with \(3x\) : \(6x - 2(3x) = 0\).

Step 3: Simplify the Equation

Simplify the equation \(6x - 6x = 0\). This reduces to \(0 = 0\), which is true for any value of \(x\).

Step 4: Substitute Back to Find y

Any value of \(x\) will satisfy both equations as long as \(y = 3x\).

Key Concepts

Substitution Method

The substitution method is a technique for solving systems of equations. Here you replace one variable in one equation with an expression derived from another equation. This can simplify the system, making it easier to solve.
In our exercise, the system is:
\begin{array}{l} \y = 3x \ \6x - 2y = 0\ \right.First, notice that the first equation gives us an expression for y in terms of x: \(y = 3x\).
By substituting this expression into the second equation, you eliminate one of the variables and simplify the solving process.

Solving Equations

Solving equations involves finding the value of the unknowns that make the equation true. Here we have two equations and two unknowns, x and y. To solve these:

• We first substituted \(y = 3x\) into the second equation: \(6x - 2(3x) = 0\).
• This simplifies to \(6x - 6x = 0\), or \(0 = 0\).

This means that no matter what value x takes, the equation will hold true. In such a case, there are infinite solutions, and instead of a single point, you get a line of solutions. For any value of x you choose, y can be found using
\(y=3x\).

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using operations like addition, subtraction, multiplication, and division. In this exercise, \(y = 3x\) is an algebraic expression for y in terms of x. Algebraic expressions are useful for representing relationships and solving equations.

When substituting \(3x\) for y in \(6x - 2y = 0\), you turn the problem into a simpler one-variable equation. This kind of manipulation shows the power of algebraic expressions in solving systems of equations efficiently. Remember, the essence of working with algebraic expressions is to make complex problems more manageable!