Problem 68 - Thinking Mathematically

Question

Explain how to solve a system of equations using the substitution method. Use \(y=3-3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

Step-by-Step Solution

Answer

The solutions are \(x=2/3\) and \(y=1\).

Step 1: Identify the Equation to Substitute

The system of equations is \(y=3-3 x\) and \(3 x+4 y=6\). The first equation is already solved for y, so this equation would be used for substitution in the second one. The equation to substitute is \(y=3-3x\).

Step 2: Substitute the first equation into the second

Next, substituting \(y\) from the first equation \(y=3-3x\) into second equation \(3x+4y=6\) gives \(3x+4(3-3x)=6\).

Step 3: Simplify the Equation

The next step is to simplify the equation. Distribute 4 to both terms in the parentheses. We get \(3x+12-12x=6\). Combine like terms to get \(-9x+12=6\).

Step 4: Solve for x

Let us solve for x by subtracting 12 from both sides of the equation. We get \(-9x= -6\). Then divide both sides of the equation by -9 and x equals \(2/3\).

Step 5: Substitute x into the first equation

Now that we have the value of x, it can be substituted into the first equation to find the value for y. Therefore, \(y= 3 - 3*(2/3)\) which simplifies to \(y = 1\).

Step 6: Verify the Solution

Finally, verification of solutions is essential to prove the correctness of the result. We substitute \(x=2/3\) and \(y=1\) into the second equation. Therefore, we would get \(3*(2/3) + 4*1 = 6\), which simplifies down to the correct equation \(2 + 4 = 6\). This verifies our solutions are correct.

Key Concepts

Substitution Method

To tackle equations with multiple variables, the substitution method stands out as a powerful tool for finding solutions efficiently. First and foremost, we aim to express one variable in terms of another using one of the given equations. In our example, the equation \(y=3-3x\) is already in the perfect format to be substituted into the other equation, allowing a single-variable equation to be formulated.

Once the substitution is made, here by replacing \(y\) in the equation \(3x+4y=6\) with \(3-3x\), we transform the system into one equation with only one variable, making it possible to find a specific value for \(x\). This seamless process eliminates the need to solve simultaneously for both variables, simplifying the problem significantly.

Simplify Equation

Simplifying an equation is akin to decluttering a room - we reduce it to its most basic form to understand it better. Following substitution, the equation often requires tidying up. Distributing, combining like terms, and isolating the variable are standard steps.

To illustrate, after substituting \(y\) in our example, we distribute \(4\) across \(3-3x\) to eliminate the parentheses, which gives us \(3x+12-12x=6\). Then we combine the \(x\)-terms, resulting in \(\-9x+12=6\). This process leaves us with a clear, solvable equation for \(x\), making it much easier to solve.

Verify Mathematical Solution

Assurance in the correctness of our solutions is paramount. Verification acts as a mathematician's seal of approval. After arriving at potential values for the variables, we plug these back into the original equations.

For our example, we check our values \(x=2/3\) and \(y=1\) with the second equation, \(3x+4y=6\). Inserting the values, we get \(3*(2/3) + 4*1 = 6\), which simplifies to \(2 + 4 = 6\), confirming our results. This step is essential, as it reassures us that the solution satisfies all the original system's conditions, certifying the solution's validity.