Problem 13 - A Graphical Approach to College Algebra

Question

Solve each system by substitution. $$\begin{aligned}x+2 y &=-1 \\\2 x+y &=4\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \), \( y = -2 \).

Step 1: Isolate one variable

Choose one of the equations to isolate a variable. In this case, isolate \( x \) from the first equation.\[ x = -1 - 2y \]

Step 2: Substitute the isolated variable

Substitute \( x = -1 - 2y \) into the second equation \( 2x + y = 4 \). This eliminates the variable \( x \).\[ 2(-1 - 2y) + y = 4 \]

Step 3: Simplify the equation

Expand and simplify the equation from Step 2:\[ -2 - 4y + y = 4 \]\[ -2 - 3y = 4 \]

Step 4: Solve for \( y \)

Solve the simplified equation for \( y \):\[ -3y = 6 \]\[ y = -2 \]

Step 5: Substitute to find \( x \)

Substitute \( y = -2 \) back into the isolated equation from Step 1: \[ x = -1 - 2(-2) \]\[ x = -1 + 4 \]\[ x = 3 \]

Step 6: Verify your solution

Plug the values \( x = 3 \) and \( y = -2 \) into the original equations to ensure they are satisfied. First equation: \( 3 + 2(-2) = -1 \) which is true. Second equation: \( 2(3) - 2 = 4 \) which is also true.

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves isolating one variable in one equation and substituting its expression in another equation. This method simplifies the problem by reducing the number of variables. To begin, choose one of the equations and express one variable in terms of the others. In our example, we isolated \( x \) from the first equation: \( x = -1 - 2y \). By doing this, we make it possible to substitute \( x \) into the second equation, eliminating one variable and making the system easier to solve.Using substitution requires some strategic thinking. You need to decide which variable to isolate and check that your substitution simplifies the process rather than complicating it. If you choose correctly, the equations become manageable and can be solved systematically.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations and expressions to reveal or isolate specific components. In the context of solving systems of equations, manipulation is a key step to simplify the problem and facilitate the solution process. In our example, after substituting \( x = -1 - 2y \) into the second equation, we expanded and simplified:

• We started with the equation \( 2(-1 - 2y) + y = 4 \).
• Next, we expanded this to \( -2 - 4y + y = 4 \).
• Simplifying further gives us \( -2 - 3y = 4 \).

This series of algebraic manipulations strips away unnecessary parts of the equations, leaving us with a clearer path to the solution. The goal is always to express the variables in a simplified form, making it easier to solve for one or more of them.

Solving Linear Equations

Solving linear equations is about finding the values of variables that satisfy all equations in a system. Once we've manipulated the equations properly, we can solve for the unknowns. In our current example, after simplifying, we had the equation \( -3y = 6 \). Solving for \( y \), we divide both sides by \(-3\):

• The solution for \( y \) is \( y = -2 \).

After finding \( y \), we substitute back into the expression we derived earlier for \( x \): \( x = -1 - 2(-2) \). This simplifies to \( x = 3 \). The process of solving these linear equations requires careful manipulation and checking to ensure each step follows logically from the last, maintaining the integrity of the solution.

Verification of Solutions

Verification of solutions is the final and essential step to confirm that the obtained values indeed satisfy the original system. After finding \( x = 3 \) and \( y = -2 \), it's important to substitute these values back into the original equations to make sure they hold true. For the first equation, substituting gives \( 3 + 2(-2) = -1 \), which is correct. For the second equation, substituting gives \( 2(3) + (-2) = 4 \), which is also correct.

• First equation: \( 3 + 2(-2) = -1 \) confirmed.
• Second equation: \( 2(3) - 2 = 4 \) confirmed.

This step is crucial because it ensures accuracy and boosts confidence in your solution. If the values satisfy all equations, you have a correct solution. If not, revisit each step to locate any errors.