Problem 44 - Introductory Algebra for College Students

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: Express One Variable in Terms of the Other

The first equation, \(y=3-3 x\), already expresses y in terms of x. Thus, \(y\) is equivalent to \(3-3 x\).

Step 2: Substitute for the Variable in the Other Equation

Substitute \(3 - 3x\) for \(y\) in the second equation, resulting in \(3 x+4 (3-3 x)=6.\)

Step 3: Simplify and Solve for the Variable

On simplifying the equation, it results in \(-9x + 12 = 6\). Solving for \(x\), yields the value \(x=2/3\).

Step 4: Substituting x into the First Equation

Substitute \(x=2/3\) into the first equation \(y=3-3 x\). This results in \(y=3 - 3(2/3)\), which simplifies to \(y=1\).