Problem 87 - Algebra and Trigonometry

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 solution to the system of equations is \(x = 2/3\) and \(y = 1\).

Step 1: Substitute for 'y' in the second equation

In the first equation, \(y = 3 - 3x\), y is isolated. This equation can be substituted into the second equation in place of y, getting \(3x + 4(3 - 3x) = 6\).

Step 2: Simplify the resulting equation

After substitution, simplify the resulting equation by doing the multiplication: \(3x + 12 - 12x = 6\). Combine like terms to simplify the equation further: \(-9x + 12 = 6\).

Step 3: Solve for 'x'

To solve for 'x', subtract 12 from both sides of the equation to isolate the term with 'x': \(-9x = 6 - 12\) which simplifies to \(-9x = -6\). Divide both sides by -9 to find 'x': \(x = -6 / -9 = 2/3\).

Step 4: Substitute 'x' into the first equation to find 'y'

Now that we have the value for 'x', substitute it into the first equation to find 'y': \(y = 3 - 3(2/3)\). After simplifying, we find that \(y = 3 - 2 = 1\).