In Exercises 5–18, solve each system by the substitution method. x + y = 4 y = 3x

Verified step by step guidance

1. Start with the given system of equations: $x+y=4y=3x$.
2. Since the second equation already expresses $y$ in terms of $x$ as $y=3x$, substitute this expression for $y$ into the first equation.
3. After substitution, the first equation becomes $x+3x=4$. Combine like terms on the left side to simplify the equation.
4. Solve the simplified equation for $x$ by isolating the variable.
5. Once you find the value of $x$, substitute it back into the expression $y=3x$ to find the corresponding value of $y$.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution is the set of values that satisfy all equations simultaneously. Understanding how to interpret and represent these systems is fundamental to solving them.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is already solved for a variable.

Solving Linear Equations

Solving linear equations means finding the value(s) of the variable(s) that make the equation true. This often involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division. Mastery of these techniques is essential for solving systems effectively.