Problem 6 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} y=2 x+3 \\ 5 y-7 x=18 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \) and \( y = 5 \).

Step 1: Identify the Substitution

The first equation is already solved for \( y \). We can substitute \( y = 2x + 3 \) into the second equation.

Step 2: Perform the Substitution

Replace \( y \) in the second equation with the expression from the first equation: \[ 5(2x + 3) - 7x = 18 \]

Step 3: Distribute and Simplify

Distribute the 5 in the equation: \[ 10x + 15 - 7x = 18 \] Simplify by combining like terms: \[ 3x + 15 = 18 \]

Step 4: Solve for x

Subtract 15 from both sides to isolate terms with \( x \):\[ 3x = 3 \] Divide by 3: \[ x = 1 \]

Step 5: Solve for y using Substituted x

Use \( x = 1 \) in the first equation to solve for \( y \):\[ y = 2(1) + 3 \] \[ y = 2 + 3 = 5 \]

Step 6: Write the Solution

The solution to the system of equations is \( x = 1 \) and \( y = 5 \).

Key Concepts

Understanding System of Equations

A system of equations consists of two or more equations that share the same set of variables. These systems can represent interconnected relationships between different quantities. The main goal in solving a system of equations is to find the values of the variables that satisfy all the equations simultaneously.

For example, let's consider the system of equations given in the exercise:

• First equation: \( y = 2x + 3 \)
• Second equation: \( 5y - 7x = 18 \)

In this system, both equations involve the variables \( x \) and \( y \). The first equation directly expresses \( y \) in terms of \( x \), which makes it ideal for substitution. The second equation involves both \( x \) and \( y \), but is not as easily disentangled. By using substitution, you can solve these equations together to uncover the values of \( x \) and \( y \) that satisfy both.

Applying Algebra in Systems

Algebra is a powerful tool that provides methods to manipulate and solve equations. The substitution method is one such algebraic tool, particularly useful in systems of equations. This method involves replacing a variable in one equation with an equivalent expression derived from another equation.

In the given problem, we have:

• \( y = 2x + 3 \)

This expression for \( y \) can be directly substituted into the second equation. Algebra helps us perform this step confidently, ensuring that our manipulation of equations leads us to correct solutions. Consider substitution like replacing a placeholder; by inserting the expression for \( y \) from the first equation into all instances of \( y \) in the second equation, you streamlines the process.

By applying algebraic principles such as distribution and combining like terms, you proceed from:

• \( 5(2x + 3) - 7x = 18 \)
• To \( 10x + 15 - 7x = 18 \)
• Further simplifying to \( 3x + 15 = 18 \)

These steps demonstrate how algebra simplifies a complex-looking system to one that's easy to solve.

Solving Equations through Substitution

Solving equations with substitution involves several logical and practical steps. The ultimate purpose is to find a solution that satisfies all equations in the system at the same time. In this problem, we use substitution to find the expressions of both \( x \) and \( y \).

After substituting \( y = 2x + 3 \) into the second equation, you simplify the math to isolate \( x \):

• Simplified to \( 3x + 15 = 18 \)
• First, subtract 15 from both sides to maintain the balance of the equation: \( 3x = 3 \)
• Then, divide both sides by 3 to solve for \( x \): \( x = 1 \)

Now that \( x \) is known, you substitute \( x = 1 \) back into the original expression for \( y \) to find:

• \( y = 2(1) + 3 = 5 \)

This approach helps ensure that the solution \( x = 1 \) and \( y = 5 \) is valid for both equations. Substitution thereby provides a structured way to untangle the values of interconnected variables.