Problem 42 - Intermediate Algebra

Question

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} r-3 t=-11 \\ r+s+t=13 \\ s-4 t=-12 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( r = 2.5 \), \( s = 6 \), \( t = 4.5 \).

Step 1: Express r from equation (1)

Given the equation \( r - 3t = -11 \), solve for \( r \). Add \( 3t \) to both sides to get: \[ r = 3t - 11 \]

Step 2: Substitute r in equation (2)

Substitute \( r = 3t - 11 \) into the second equation \( r + s + t = 13 \). \[ (3t - 11) + s + t = 13 \] Simplify the equation to \( 4t + s = 24 \).

Step 3: Use equation (3)

The third equation is \( s - 4t = -12 \). You can now write it as \( s = 4t - 12 \).

Step 4: Create a new system of equations

From steps 2 and 3, we have two expressions for \( s \): \( 4t + s = 24 \) and \( s = 4t - 12 \). Set these equal to solve for \( t \).

Step 5: Solve for t

Set the equations for \( s \) equal: \[ 4t + (4t - 12) = 24 \] Simplify to get: \[ 8t - 12 = 24 \] Add 12 to both sides: \[ 8t = 36 \] Divide by 8: \[ t = \frac{36}{8} = 4.5 \]

Step 6: Solve for r using t

Now that you know \( t = 4.5 \), substitute it back into the expression for \( r \): \[ r = 3(4.5) - 11 \] Simplify: \[ r = 13.5 - 11 = 2.5 \]

Step 7: Solve for s using t

Substitute \( t = 4.5 \) into the equation for \( s \): \[ s = 4(4.5) - 12 \] Simplify: \[ s = 18 - 12 = 6 \]

Step 8: Verify the solution in all equations

Substitute \( r = 2.5 \), \( s = 6 \), and \( t = 4.5 \) into the original equations to verify.- Equation 1: \( r - 3t = 2.5 - 13.5 = -11 \), check,- Equation 2: \( r + s + t = 2.5 + 6 + 4.5 = 13 \), check,- Equation 3: \( s - 4t = 6 - 18 = -12 \), check.

Key Concepts

Understanding the Substitution Method

The substitution method is a technique used to solve a system of linear equations. Its core idea is to solve for one variable and then substitute that expression into the other equations. This method systematically reduces a system of equations to a simpler form, allowing us to solve it step-by-step. Let's break it down:

• First, choose one equation and solve for one of its variables. For example, in the equation \( r - 3t = -11 \), solve for \( r \) to get \( r = 3t - 11 \).
• Next, substitute this expression into the other equations. Replace \( r \) in another equation with \( 3t - 11 \). This reduces the number of variables, making it easier to solve.
• Continue to substitute expressed variables into other equations until all variables are isolated. This might involve solving for one variable in terms of another and substituting into the remaining equations.

Using the substitution method can be particularly beneficial when equations involve interdependent variables, providing a clear path to find all unknowns.

Introduction to Linear Equations

Linear equations are fundamental components of algebra that form the basis of many mathematical models. A linear equation is an equation where the highest power of the variable is one. They graphically represent straight lines and are expressed in the form \( ax + by + cz = d \).

Understanding their properties is crucial for solving them effectively:

• A linear equation in two variables (like \( x \) and \( y \)) has a solution set that forms a line on a two-dimensional plane.
• In three variables, like our system with \( r, s, \) and \( t \), the solution set typically forms a plane in three-dimensional space.
• They can either represent parallel lines, coinciding lines, or intersecting lines/planes, depending on their relationships.

When solving, employing algebraic techniques like substitution or elimination simplifies the objective to finding where these lines intersect (common solutions). The ease of manipulation and the visual representation make linear equations vital in mathematical problem-solving.

The Nature of Consistent and Independent Systems

When dealing with systems of linear equations, terms like 'consistent' and 'independent' arise frequently to describe the nature of the solutions. Let's dive into what these terms mean.

• Consistent systems: A system of equations is consistent if there is at least one set of values for the variables that satisfies all the equations simultaneously. Our original system is consistent because we found specific values for \( r, s, \) and \( t \) that make all three equations true.
• Independent systems: An independent system means that each equation in the system provides unique information. For instance, no equation is a multiple of the other, which results in exactly one solution for the system. Our system is independent, as each equation contributes necessary information and leads us to a single solution.

If a system is not consistent or independent, it may be inconsistent (no solutions) or dependent (infinite solutions due to equations being multiples of each other). Recognizing these properties guides the strategy used to approach solving a system.