Problem 70 - College Algebra

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 y in the second equation

Since we have \(y = 3 - 3x\), we can substitute y into the second equation \(3x + 4y = 6\). It becomes \(3x + 4(3 - 3x) = 6\).

Step 2: Simplify the equation

Distributing the 4 through the parentheses \(3x + 12 - 12x = 6\), simplify the equation into \(-9x + 12 = 6\).

Step 3: Solve for x

Subtract 12 from both sides, which gives us \(-9x = -6\). Then, divide by -9 to solve for x. We find that \(x = 2/3\).

Step 4: Substitute x in the first equation

Substitute \(x = 2/3\) into the first equation \(y = 3 - 3x\). This becomes \(y = 3 - 3(2/3) = 1\).

Step 5: Solution of the system of equations

From the previous steps, the solution of the system of equations is the ordered pair \((2/3, 1)\). This means x equals to 2/3 and y equals to 1.