Problem 16 - Algebra 2

Question

Solve each system of linear equations by graphing. \(2 x+3 y=7\) \(2 x-3 y=7\)

Step-by-Step Solution

Answer

The solution is \( \left(\frac{7}{2}, 0\right) \).

Step 1: Add the equations

Adding: \( (2x+3y) + (2x-3y) = 7+7 \Rightarrow 4x = 14 \Rightarrow x = 7/2 \).

Step 2: Find y

Substitute into first equation: \( 2(7/2) + 3y = 7 \Rightarrow 7 + 3y = 7 \Rightarrow y = 0 \).

Step 3: State solution

The solution is \( (x, y) = \left(\frac{7}{2}, 0\right) \) or \( (3.5, 0) \).

Key Concepts

Graphing Linear Equations

Graphing linear equations is a method used to visually display the solutions of linear equations on a coordinate plane. To graph a linear equation,

• identify the equation you need to plot,
• rewrite the equation in the slope-intercept form, which makes it easier to understand and graph,
• determine the y-intercept and slope from the slope-intercept form,
• plot the y-intercept on the y-axis,
• use the slope to find another point on the graph,
• draw a straight line through the points, extending it in both directions.

For example, for the equation \(y = -\frac{2}{3}x + \frac{7}{3}\), the line will cross the y-axis at \(\frac{7}{3}\). The slope of \(-\frac{2}{3}\) tells you to go down 2 units for every 3 units you move to the right. Repeat this process for the second equation \(y = \frac{2}{3}x - \frac{7}{3}\), and place both lines on the same graph to explore their intersections.

Slope-Intercept Form

The slope-intercept form is one of the most straightforward forms of a linear equation. It is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, showing the rate at which the line ascends or descends. The coefficient \(b\) indicates the y-intercept, which is where the line crosses the y-axis. This form is particularly useful:

• because it provides direct information about the slope and the y-intercept,
• makes graphing simple by identifying the starting point and the direction of the line,
• facilitates comparison between multiple equations by showing differences in their slopes and intercepts.

By converting our original equations to slope-intercept form, they become easier to graph and compare. The lines resulting from \(y = -\frac{2}{3}x + \frac{7}{3}\) and \(y = \frac{2}{3}x - \frac{7}{3}\) allow us to see their intercepts and slopes clearly.

Solving Systems of Equations

Solving systems of equations involves finding the values of the variables that satisfy all equations in the system simultaneously. One effective method is graphing:

• First, ensure each equation is in a graph-friendly form, like slope-intercept form.
• Next, display each line graphically on the same coordinate plane.
• The solution to the system is the point where the graphs intersect.

If the lines intersect at a single point, that point is the unique solution to the system. For example, graphing \(y = -\frac{2}{3}x + \frac{7}{3}\) and \(y = \frac{2}{3}x - \frac{7}{3}\) reveals that these lines meet at exactly one point. This intersection represents the set of values that solves both equations, providing us with the solution to the system.

Linear Equations in Two Variables

Linear equations in two variables are expressions that describe a relationship between two different quantities. These equations typically have the general form of \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables.

• Each linear equation in two variables represents a straight line when graphed on a coordinate plane.
• The solutions to these equations are ordered pairs \((x, y)\) that satisfy the equation.
• These equations are fundamental in studying algebra, as they lay the foundation for more complex mathematical concepts.
• Understanding how these equations work and how they can be manipulated is key in problem-solving scenarios involving multiple linear relationships.

In our example, the two equations \(2x + 3y = 7\) and \(2x - 3y = 7\) are linear equations in two variables, indicating two different lines on a plane. Solving them involves finding intersection points, thereby understanding the relationships represented by these lines.