Problem 105 - College Algebra with Modeling and Visualization

Question

Solve the system of linear equations $$(a) graphically,\quad(b) numerically,\quad and\quad(c) symbolically.$$ $$\begin{array}{r}-2 x+y=0 \\\7 x-2 y=3\end{array}$$

Step-by-Step Solution

Answer

The solution is (1, 2).

Step 1: Graphical Solution

To solve the system graphically, plot both equations on a coordinate plane.For the first equation, \(-2x + y = 0\) or \(y = 2x\):- A point is (0,0) because if \(x = 0\), then \(y = 0\).- Another point can be \(x = 1\), then \(y = 2(1) = 2\), giving (1,2).For the second equation, \(7x - 2y = 3\):- Rewrite to find points: when \(x = 0\), \(-2y = 3\) gives \(y = -\frac{3}{2}\), point (0, -\frac{3}{2}).- When \(y = 0\), \(7x = 3\) gives \(x = \frac{3}{7}\), point (\frac{3}{7}, 0).Draw both lines on a graph to find the intersection, approximately (1,2).

Step 2: Numeric Approximation

Using substitution or elimination, we find decimal or fractional solutions.Using substitution, substitute \(y = 2x\) into the second equation:\[7x - 2(2x) = 3\]\[7x - 4x = 3\]\[3x = 3\]\[x = 1\]Substitute \(x = 1\) back into \(y = 2x\):\[y = 2(1) = 2\]Numerically, the solution is (1, 2).

Step 3: Symbolic Verification

Solving symbolically involves using algebraic manipulation.Start by expressing one of the equations in terms of one variable, say \(y\):From \(-2x + y = 0\), \(y = 2x\).Substitute into the second equation:\[7x - 2(2x) = 3\]\[7x - 4x = 3\]\[3x = 3\]\[x = 1\]Substitute \(x = 1\) back into the first equation:\[y = 2(1) = 2\]Thus, the symbolic solution is \((x, y) = (1, 2)\).

Key Concepts

Graphical Solution

When tackling a system of linear equations graphically, we translate each equation into a line on a coordinate plane. This method allows us to visually identify the point where they intersect, representing the solution to the system.

To plot each line, we first need to convert the equations into a form that's easy to graph. Consider the set of equations:

• Equation 1: \(-2x + y = 0\)
• Equation 2: \(7x - 2y = 3\)

Starting with the first equation, \(y = 2x\), we can easily graph by selecting points such as \(x = 0, y = 0\) and \(x = 1, y = 2\). These points give us a clear line when plotted.

The second equation needs rearranging for better clarity. Solving for \(y\) in terms of \(x\), we find points like \(x = 0, y = -\frac{3}{2}\) and \(x = \frac{3}{7}, y = 0\). Plotting these gives another line.

The intersection of these two lines on the graph is the solution to the system. Inspecting closely, this intersection is at \(x = 1\) and \(y = 2\), solving the system visually.

Numeric Approximation

Numeric approximation involves solving equations using numbers, often involving methods like substitution or elimination.

• Substitution involves solving an equation for one variable and plugging it into another.
• Elimination focuses on eliminating one variable by adding or subtracting equations.

For our system, substitution is straightforward:
1. From \(-2x + y = 0\), solve for \(y\) as \(y = 2x\).
2. Substitute \(y = 2x\) into the second equation \(7x - 2y = 3\): \(7x - 2(2x) = 3\) which simplifies to \(3x = 3\). Solving gives \(x = 1\).
3. Insert \(x = 1\) back into \(y = 2x\), resulting in \(y = 2\).

Thus, through numeric approximation, we find the solution to be \[ (x, y) = (1, 2) \], aligning with our earlier observation from the graph.

Symbolic Verification

Symbolic verification is ideal to confirm solutions algebraically, ensuring our results are accurate across different methods.

Here, algebraic manipulation is key:

• Start with \(-2x + y = 0\), solved to \(y = 2x\).
• Substitute \(y = 2x\) into the other equation, \(7x - 2y = 3\), simplifying to \(3x = 3\).
• Solve \(3x = 3\) to find \(x = 1\).
• Insert \(x = 1\) back into the equation for \(y\) to confirm \(y = 2\).

This symbolic approach affirms the solution as \[ (x, y) = (1, 2) \], confirming accuracy found earlier both graphically and numerically.