Problem 125 - Elementary Algebra

Question

Solve the system of equations \(\left\\{\begin{array}{l}3 x+y=12 \\\ x=y-8\end{array}\right.\) by substitution and explain all your steps in words.

Step-by-Step Solution

Answer

x = 1, y = 9

Step 1 - Solve the second equation for x

The second equation is given by \(x = y - 8\). This equation is already solved for \(x\), so we can use this expression to substitute \(x\) in the first equation.

Step 2 - Substitute x in the first equation

Substitute \(x = y - 8\) into the first equation \(3x + y = 12\). This gives us \(3(y - 8) + y = 12\).

Step 3 - Simplify and solve for y

Expand and simplify the equation: \[3(y - 8) + y = 12\] \[3y - 24 + y = 12\] \[4y - 24 = 12\] \[4y = 36\] \[y = 9\]. So, \(y = 9\).

Step 4 - Substitute y back into the second equation to find x

Substitute \(y = 9\) into the second equation \(x = y - 8\): \(x = 9 - 8\), so \(x = 1\).

Step 5 - Write the solution

The solution to the system of equations is \(x = 1\) and \(y = 9\).

Key Concepts

substitution method

In solving a system of linear equations, the substitution method is a key approach. It involves solving one of the equations for one variable and then substituting this value into the other equation. This helps to reduce the system to a single equation with one variable, which can be solved more easily.

Let's break it down step by step with our given problem:

• The original system is \(\begin{array}{l} 3x + y = 12 \ x = y - 8 \end{array}\).
  
• The second equation \(x = y - 8\) is already solved for \(x\).

This makes our job easier, as we have directly obtained an expression for \(x\) in terms of \(y\).
• Our next step is to substitute this expression (\(y - 8\)) for \(x\) in the first equation (\

algebraic solution steps

Algebraic steps are essential for systematically solving equations and ensuring accuracy. Let's revisit the exact steps for our problem:

• Step 1: Solve the second equation for \(x\): \(x = y - 8\). This step is quite straightforward since the equation is already solved for \(x\).
• Step 2: Substitute \(x\) in the first equation: Replace \(x\) with \(y - 8\) in \(3x + y = 12\), giving us \(3(y - 8) + y = 12\).
• Step 3: Simplify and solve for \(y\): Expand and combine like terms: \[3(y - 8) + y = 12\] simplifies to \[4y - 24 = 12\], and then \[4y = 36\]. Finally, divide by 4 to get \[y = 9\].


• Step 4: Substitute \(y\) back into the second equation to find \(x\): With \(y = 9\), substitute into \(x = y - 8\), yielding \(x = 1\).
• Step 5: Write the solution: The final solution is \(x = 1\) and \(y = 9\).

These steps illustrate how to systematically tackle the problem through substitution, simplification, and solving for variables.