91Ó°ÊÓ

A system of equations consists of two or more equations that share common variables. When solving such systems, we aim to find values for the variables that satisfy all equations simultaneously. In our example, the system includes the variables \(x\) and \(y\) in the equations:

• \(2x + y = 7\)
• \(4x - 3y = -16\)

The elimination method is a systematic technique for solving systems of equations. The goal is to eliminate one variable so we can easily solve for the other.
Here’s a step-by-step explanation:1. **Align both equations:** Start by writing both equations one below the other. \text{Step 1:} \begin{aligned} 2x + y &= 7 \ 4x - 3y &= -16 2. **Eliminate a variable:** Choose a variable to eliminate by making the coefficients (numbers in front of the variable) in both equations equal but opposite. In this case, we will eliminate \(y\). To achieve this, multiply the first equation by 3: \text{Step 2:} \begin{aligned} 3 \times (2x + y) &= 3 \times 7 \ 6x + 3y &= 21 3. **Add the equations:** Add the modified first equation to the second equation to eliminate \(y\). \text{Step 3:} \begin{aligned} (6x + 3y) + (4x - 3y) &= 21 + (-16) \ 10x &= 5 4. **Solve for \(x\):** Divide both sides of the resulting equation by 10: \text{Step 4:} \begin{aligned} x &= \frac{5}{10} \ x &= \frac{1}{2}

Once we have a solution, it's crucial to verify it by substituting the values back into the original equations. This ensures accuracy and confirms that the solution works for all equations.
1. Substitute \(x = \frac{1}{2}\) back into one of the original equations to find \(y\): \text{Step 5:} \begin{aligned} 2(\frac{1}{2}) + y &= 7 \ y &= 7 - 1 \ y &= 6 2. Verify by checking both values (\(x = \frac{1}{2}\) and \(y = 6\)) in the second equation: \text{Step 6:} \begin{aligned} 4(\frac{1}{2}) - 3(6) &= -16 \ 2 - 18 &= -16

Algebraic manipulation involves using arithmetic operations (addition, subtraction, multiplication, division) to rearrange equations and solve for unknown variables. Learning these skills is essential for solving systems of equations efficiently.
Key steps involved in manipulation for the elimination method are:

• **Multiplication to match coefficients**: Adjust equations so that one variable has the same coefficient in both equations.
  \text{Example: multiplying \(2x + y = 7\) by 3 to get \(6x + 3y = 21\)}.
• **Addition/Subtraction to eliminate a variable**: Combine equations to cancel out one variable.
  \text{Example: \(6x + 3y + 4x - 3y = 21 - 16\) results in \(10x = 5\)}.
• **Solving for the remaining variable**: Use basic algebra to solve for the single variable left in the equation.
  \text{Example: \(10x = 5\) leads to \(x = \frac{1}{2}\)}.