Problem 87 - College Algebra Essentials

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 \((x, y) = (2/3, 1)\).

Step 1: Identify the variable to substitute

The first equation is \(y=3-3 x\), which has already isolated y. Therefore, the substitution will replace y in the second equation with \(3-3x\).

Step 2: Substitute the expression into the second equation

Substitute y in the second equation \(3 x + 4 y = 6\) with the expression from the first equation. The second equation becomes \(3 x + 4 (3-3x) = 6\).

Step 3: Simplify the equation

Expand and simplify the equation: \(3 x + 12 -12x = 6\), which reduces to \(-9x + 12 = 6\).

Step 4: Solve for x

Isolate x by subtracting 12 from both sides and then dividing by -9. This gives \(x = 2/3\).

Step 5: Substitute x into the first equation

Substitute \(x = 2/3\) into the first equation to solve for y: \(y = 3 - 3 * (2/3)\), which simplifies to \(y = 1\).

Step 6: State the solution to the system

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