Problem 104 - College Algebra with Modeling and Visualization

Question

Solve the system of linear equations $$(a) graphically,\quad (b) numerically,\quad and\quad (c) symbolically.$$ $$ \begin{array}{l} 3 x+2 y=-2 \\ 2 x-y=-6 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \), \( y = 2 \).

Step 1: Graph Each Equation

To solve the system graphically, start by converting each equation to slope-intercept form (i.e., \( y = mx + b \)).For the first equation \( 3x + 2y = -2 \), solve for \( y \):\[ y = -\frac{3}{2}x - 1 \]For the second equation \( 2x - y = -6 \), solve for \( y \):\[ y = 2x + 6 \]Plot both lines on a coordinate plane, identifying intercepts and slopes accordingly.

Step 2: Identify Intersection Point on Graph

Observe where the two lines intersect on the graph. This intersection represents the solution to the system of equations, as both equations are satisfied at this point. Estimate the coordinates of the intersection.

Step 3: Set Up Numerical Solution

Next, we'll solve numerically by using matrices or substitution. For substitution, from the second equation \( y = 2x + 6 \), substitute \( y \) in the first equation:\[ 3x + 2(2x + 6) = -2 \]Expand and simplify this equation.

Step 4: Solve the Simplified Equation

Simplify the expression:\[ 3x + 4x + 12 = -2 \]Combine like terms:\[ 7x = -14 \]Solve for \( x \):\[ x = -2 \]

Step 5: Substitute Back to Find y Numerically

Substitute \( x = -2 \) back into one of the original equations to find \( y \). We use \( y = 2x + 6 \):\[ y = 2(-2) + 6 = -4 + 6 \]Thus, \( y = 2 \). The numerical solution is \( (x, y) = (-2, 2) \).

Step 6: Solve System Symbolically Using Substitution

To verify symbolically, start with substituting \( y = 2x + 6 \) into \( 3x + 2y = -2 \):\[ 3x + 2(2x + 6) = -2 \]This gives \[ 3x + 4x + 12 = -2 \], leading to \( x = -2 \) as previously solved.

Step 7: Verify Symbolical Solution with Second Equation

Substitute \( x = -2 \) into the second equation \( 2x - y = -6 \):\[ 2(-2) - y = -6 \]This results in \( -4 - y = -6 \). Solving for \( y \), we find \( y = 2 \). Thus, solution \( (x, y) = (-2, 2) \) satisfies both equations.

Key Concepts

graphical solution

To solve a system of linear equations graphically, you need to plot each equation on the same coordinate plane and look for the point where the two lines intersect. This intersection point provides the solution to the system, as it is where both equations hold true simultaneously.

Start by rewriting each equation in slope-intercept form, which is given by \( y = mx + b \) where \( m \) represents the slope and \( b \) the y-intercept. For the example equations:

• First equation: \( 3x + 2y = -2 \) becomes \( y = -\frac{3}{2}x - 1 \)
• Second equation: \( 2x - y = -6 \) becomes \( y = 2x + 6 \)

Plot both lines by identifying the intercepts and slopes. Look at the graph to determine where the two lines intersect. This point is \( (-2, 2) \) for these particular equations.

The graphical method is a great way to visually understand solutions, especially in simpler systems. It's essential, however, to plot accurately to identify the correct intersection point.

numerical methods

Numerical methods for solving systems of equations involve computational techniques to find solutions. For linear equations, this could mean using substitution or matrix-based methods. Here, let’s focus on substitution, a straightforward approach for smaller systems.

To apply substitution, solve one of the equations for one of the variables and substitute this expression into the other equation. In the given system:

• Take the second equation \( y = 2x + 6 \)
• Substitute \( y \) in the first equation: \( 3x + 2(2x + 6) = -2 \)

Simplify the equation to find \( x \). Combine like terms to get \( 7x = -14 \), leading to \( x = -2 \). Next, substitute \( x = -2 \) back into one of the original equations, which helps to find \( y \):

• Using \( y = 2x + 6 \), we find \( y = 2 \).

Thus, the numerical solution also gives us \( (-2, 2) \). Substitution is often sufficient for systems with straightforward computations, making it a handy method for students.

symbolic solution

Solving systems symbolically often involves algebraic methods that are more general, like substitution or elimination, and provides exact solutions. To solve our system symbolically using substitution, begin similarly to the numerical example:

• Express \( y \) in terms of \( x \) from one equation, such as \( y = 2x + 6 \)
• Substitute this expression into the other equation: \( 3x + 2(2x + 6) = -2 \)

Simplifying this equation, you get \( 7x + 12 = -2 \), leading to the solution \( x = -2 \).

To verify, substitute \( x = -2 \) back into both original equations to ensure it satisfies them:

• In the first equation: \( 3(-2) + 2(2) = -2 \) checks out
• In the second equation: \( 2(-2) - 2 = -6 \) verifies correctly

This confirms the solution \( (-2, 2) \) robustly satisfies the system symbolically. Symbolic methods are reliable for deriving accurate results and are particularly insightful for exploring algebraic relationships in equations.