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Mini-Investigation Consider the system $$ \left\\{\begin{array}{l} 3 x+2 y=7 \\ 2 x-y=4 \end{array}\right. $$ a. Solve each equation for \(y\) and graph the result on your calculator. Sketch the graph on your paper. b. Add the two original equations and solve the resulting equation for \(y\). Add this graph to your graph from \(12 \mathrm{a}\). What do you notice? c. Multiply the second original equation by 2 , then add this to the first equation. Solve this equation for \(x\) and add its graph to your graph from \(12 \mathrm{a}\). What do you notice? d. Multiply the first original equation by 2 and the second by \(-3\), then add the results. Solve this equation for \(y\) and add its graph to \(12 \mathrm{a}\). What do you notice? e. What is the solution to the system of equations? How does this point relate to the graphs you drew in \(12 a-d\) ? f. Write a few sentences summarizing any conjectures you can make based on this exercise.

Step by step solution

01

Solve each equation for y

We start by solving both equations in terms of y. For the first equation, \(3x + 2y = 7\):\[ 2y = 7 - 3x \]\[ y = \frac{7 - 3x}{2} \]For the second equation, \(2x - y = 4\):\[ -y = 4 - 2x \]\[ y = 2x - 4 \]Graph these two linear equations as \(y = \frac{7 - 3x}{2}\) and \(y = 2x - 4\).

02

Add the original equations and solve for y

Add the two original equations:\[(3x + 2y) + (2x - y) = 7 + 4\]\[5x + y = 11\]Solve for y:\[y = 11 - 5x\]Graph this equation on the same axes. Notice where this line intersects with the graphs obtained in Step 1.

03

Multiply and add equations to solve for x

Multiply the second equation by 2 and add to the first equation:\[(2)(2x - y) = 2(4)\]\[4x - 2y = 8\]Now add this to the first equation:\[(3x + 2y) + (4x - 2y) = 7 + 8\]\[7x = 15\]Solve for x:\[x = \frac{15}{7}\]Graph the vertical line \(x = \frac{15}{7}\) on the same axes. Notice the intersection point.

04

Multiply and add equations to solve for y

Multiply the first equation by 2 and the second equation by -3:\[(2)(3x + 2y) = 2(7)\]\[6x + 4y = 14\]\[(-3)(2x - y) = -3(4)\]\[-6x + 3y = -12\]Add these:\[(6x + 4y) + (-6x + 3y) = 14 + (-12)\]\[7y = 2\]Solve for y:\[y = \frac{2}{7}\]Graph the horizontal line \(y = \frac{2}{7}\) on the same axes.

05

Find the solution to the system of equations

The solution to the system of equations is the point where all the lines from previous steps intersect. From the previous calculations, \(x = \frac{15}{7}\) and \(y = \frac{2}{7}\).

06

Summarize findings

The intersection point found in all steps, \(\left(\frac{15}{7}, \frac{2}{7}\right)\), represents the solution to the system of equations. This point lies at the intersection of all graphs plotted. This exercise demonstrates how manipulation of equations maintains the unique solution, illustrating consistency in linear equations through different algebraic manipulations and graphical interpretations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Linear Equations

Graphing linear equations is a useful skill in mathematics, as it allows us to visualize the relationships between variables. Each linear equation in two variables, such as the ones in our exercise, represents a straight line on a graph.
To graph a linear equation, you'll first want to solve it for one of the variables, usually y. This results in an expression such as \( y = mx + b \), where m represents the slope of the line and b is the y-intercept. These characteristics give you the angle and starting point of your line on a coordinate plane.
You plot the y-intercept first, then use the slope (rise over run) to determine the next points, drawing a line through them. Using a graphing calculator can streamline visualizing multiple equations simultaneously, as it automatically scales and delineates these lines accurately on a grid.

Solving Equations Graphically

Solving equations graphically involves finding points where graphs (lines) intersect, which correspond to solutions of the equations. For a system of linear equations, each line represents a possible solution set.
When graphs are plotted on the same axes, the intersection points show values of variables that satisfy all equations simultaneously. This graphical solution method is intuitive and provides a clear visual depiction of solutions.
With our equations in the exercise, solving graphically revealed the intersection point of multiple lines. This point is consistent with our algebraic solutions, reinforcing the relationship between algebra and geometry in solving problems.

Intersection of Lines

The intersection of lines in a graph is critical when solving systems of linear equations, as it signifies the solution(s) common to the equations involved.
Through graphical plotting, the intersection point tells us where both conditions laid out by the equations are satisfied. If two lines meet at a single point, it means this point is the unique solution to the system.
In our exercise, as we plotted each derived line, we observed a common intersection point at \( ( \frac{15}{7}, \frac{2}{7} ) \). This means that these coordinate values simultaneously satisfy both equations in the system. When using graphical methods, spotting such intersections is essential for verifying algebraic calculations.

Algebraic Manipulation

Algebraic manipulation is a powerful tool in mathematics, allowing us to transform equations into various forms to extract solutions or simplify calculations.
In our exercise, we used several algebraic methods. First, we isolated variables to form easier-to-graph linear equations. Then, we combined equations through addition or multiplication to eliminate variables systematically, revealing either x or y values.
Each manipulation was designed to either simplify graphing or directly solve for a variable, leading us to the point of intersection \( ( \frac{15}{7}, \frac{2}{7} ) \). This demonstrates how different forms of algebra can yield the same solution, showing the consistency and reliability of these operations in solving linear equations.