Problem 47 - Elementary and Intermediate Algebra

Question

Solve using substitution: $$\left\\{\begin{array}{l}x+y=4 \\\y=3 x\end{array}\right.$$

Step-by-Step Solution

Answer

\( x = 1 \), \( y = 3 \)

Step 1: Identify Equations

We are given a system of two equations: 1. \( x + y = 4 \) 2. \( y = 3x \)

Step 2: Substitute y in the First Equation

Since the second equation gives us \( y = 3x \), we can substitute \( 3x \) for \( y \) in the first equation: \( x + 3x = 4 \)

Step 3: Simplify the Equation

Combine like terms in the equation: \( 4x = 4 \)

Step 4: Solve for x

Divide both sides by 4 to isolate \( x \): \( x = \frac{4}{4} \) \( x = 1 \)

Step 5: Substitute x back into y=3x

Now that we have \( x = 1 \), substitute it back into the equation \( y = 3x \): \( y = 3(1) \) \( y = 3 \)

Key Concepts

Solving Linear Equations

Linear equations are one of the most straightforward equations in algebra. A linear equation has no variables with exponents greater than one, and they often appear in the form: \( ax + b = c \). Solving these equations typically involves isolating the variable on one side of the equation. Here's how you can approach solving a basic linear equation:

• Identify the linear equation in its standard form.
• Use basic arithmetic operations to isolate the variable.
• If needed, simplify by combining like terms.
• Perform the same operation on both sides of the equation to maintain equality.

For example, if you have an equation like \( x + 3 = 7 \), you would subtract 3 from both sides to find \( x = 4 \). Linear equations form the foundation for solving more complex systems that involve multiple equations, such as in the substitution method.

Systems of Equations

A system of equations consists of multiple equations that you deal with at the same time. These equations can be linear, and they intersect at the same point on a graph. For example, in this problem, we have the system:

• \( x + y = 4 \)
• \( y = 3x \)

Solving a system of equations means finding all variable values that satisfy each equation simultaneously. If you graph these equations, they would intersect at the point \( (x, y) \). There are different methods to solve systems of equations, such as substitution and elimination. Substitution is particularly useful when one equation is already solved for a variable, simplifying the process of plugging it into another equation.

Algebraic Substitution

The substitution method is a powerful algebraic tool for solving systems of equations. It involves replacing one variable in an equation with another expression that is equal to the variable, making the equation easier to solve. Here's a step-by-step guide on how to use substitution:

• Solve one equation for one variable, if necessary.
• Take that expression and substitute it into the other equation.
• Solve the newly formed equation for its single variable.
• Use the known variable value to solve for the remaining variable in one of the original equations.

In the system of equations provided, since \( y = 3x \) directly defines \( y \) in terms of \( x \), you substitute \( 3x \) in place of \( y \) in the first equation \( x + y = 4 \). This substitution leads to a single-variable equation \( 4x = 4 \), which is simple to solve. Once \( x \) is found, substitute it back to find \( y \). This entire process highlights the elegance and efficiency of algebraic substitution in systems of equations.