Problem 31 - Elementary and Intermediate Algebra

Question

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} a \quad \quad -3 c=6 ,\\\ b+2 c =2 ,\\\ 7 a-3 b-5 c =14 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( a = \frac{22}{7}\), \( b = 6\), \( c = -2 \). All equations hold true.

Step 1: Solve for one variable

Start by solving one of the simplest equations for one of the variables. From the equation \begin{aligned}-3c=6to solve for c,divide both sides by -3\[ c = \frac{6}{-3} \] Thus, \[ c = -2 \]

Step 2: Substitute the solved variable

Substitute the value of c into the second equation:b + 2*(-2) = 2,which simplifies tob - 4 = 2 To solve for b,add 4 to both sides:\[ b = 6 \]

Step 3: Substitute the variables into the third equation

Use the values of b and c in the third equation to solve for a.Substitute b = 6 and c = -2 into7a - 3b - 5c = 14,7a - 3(6) - 5(-2) = 14 simplifies to 7a - 18 + 10 = 14,which further simplifies to7a - 8 = 14.Adding 8 to both sides:\[ 7a = 22 \]Now, divide by 7:\[ a = \frac{22}{7} \]

Step 4: Verify the solution

Verify the solution by substituting \( a = \frac{22}{7}, b = 6, c = -2 \) into the original equations to ensure they hold true:-3(-2) = 6: ✔️ Correct 6 + 2(-2) = 2 6 - 4 = 2: ✔️ Correct7(\frac{22}{7}) - 3(6) - 5(-2) = 1422 - 18 + 10 = 14: ✔️ Correct

Key Concepts

dependent equations

Dependent equations are systems where all equations describe the same line or plane. In other words, they are essentially the same equation, just in different forms. This means there are an infinite number of solutions because any point on the line or plane satisfies all equations. Detecting dependent equations involves checking if any equation can be manipulated (by multiplication, division, addition, or subtraction) to resemble another equation in the system. In our exercise, none of the given equations is a simple multiple or combination of the others, so they are not dependent. Therefore, solving the system should give us a unique solution, assuming they intersect at a single point.

substitution method

The substitution method involves solving one of the equations for one variable and substituting that expression into the other equation(s). This reduces the system to a simpler form, often a single variable equation. Here’s a step-by-step guide to using the substitution method:

• Solve one equation for one variable.
• Substitute this expression into the other equation(s) to find the values of the other variables.
• Substitute the found values back into the initial solved expression to find the remaining variable.

In our exercise, we:

• Solved -3c = 6 to find c = -2.
• Substituted c = -2 into b + 2c = 2 to find b.
• Substituted b = 6 and c = -2 into 7a - 3b - 5c = 14 to find a.

Finally, we verified the solution by plugging the values of a, b, and c back into the original equations to ensure they were satisfied.

linear equations

Linear equations are equations of the first degree, meaning they have no exponents higher than 1. They can be represented in forms like ax + by = c, where a, b, and c are constants. Solving systems of linear equations often involves finding a common point of intersection. There are three possible outcomes when solving linear equations:

• A single solution (the lines intersect at one point).
• No solution (the lines are parallel and never intersect).
• Infinite solutions (the lines are dependent and overlap entirely).

For our exercise, each step confirmed scenarios that depend on the type of system:

• Solved individual equations for one variable.
• Substituted to find other variables.
• Verified consistency across all original equations.

Through these steps, we identified a unique solution without dependent equations or contradictions, meaning the system had a single solution (a specific intersection point).