Problem 28 - Algebra Form and Function

Question

Solve the system of equations graphically. $$ \left\\{\begin{array}{l} y=22+4(x-8) \\ y=11-2(x+6) \end{array}\right. $$

Step-by-Step Solution

Answer

Question: What is the solution to the given system of equations: $$ y=22+4(x-8) \\ y=11-2(x+6) $$ Answer: The solution to the given system of equations is \((\frac{3}{2}, -4)\).

Step 1: 1. Rewrite the equations in slope-intercept form (y=mx+b)

The given equations in the system are: $$ y=22+4(x-8) \\ y=11-2(x+6) $$ We can rewrite them in the form y = mx + b, by simplifying the expressions: For the first equation: $$ y = 22 + 4(x-8) \\ y = 22 + 4x - 32 \\ y = 4x - 10 $$ For the second equation: $$ y = 11 - 2(x + 6) \\ y = 11 - 2x - 12 \\ y = -2x - 1 $$ So, the simplified equations are: $$ \left\\{\begin{array}{l} y = 4x - 10 \\ y = -2x - 1 \end{array}\right. $$

Step 2: 2. Identify the slopes and y-intercepts

For each equation, we can identify the slope (m) and y-intercept (b) using their respective slope-intercept form: For the first equation (\(y = 4x - 10\)), the slope is 4, and the y-intercept is -10. For the second equation (\(y = -2x - 1\)), the slope is -2, and the y-intercept is -1.

Step 3: 3. Sketch the lines on the coordinate plane

Using the slopes and y-intercepts, we can plot the lines on a coordinate plane: 1. Plot the y-intercept of the first line (-10) on the y-axis. 2. Starting from this point, move up by 4 units and then 1 unit to the right to plot another point due to the slope of 4 (rise = 4, run = 1). 3. Repeat this process to draw the first line. 4. Plot the y-intercept of the second line (-1) on the y-axis. 5. Starting from this point, move down 2 units then 1 unit to the right to plot another point due to the slope of -2 (rise = -2, run = 1). 6. Repeat this process to draw the second line. The point where both lines intersect on the graph will be our solution.

Step 4: 4. Find the point of intersection

By observing the graph, we identify the point where both lines intersect. Alternatively, we can solve these equations step-by-step by equating the y's to each other. $$ 4x - 10 = -2x -1 \\ 4x + 2x = -1 + 10 \\ 6x = 9 \\ x = \frac{9}{6} = \frac{3}{2} $$ Now that we have the x-coordinate, plug it into any of the two equations to find the y-coordinate. We'll use the first equation: $$ y = 4x - 10 \\ y = 4(\frac{3}{2}) - 10 \\ y = 6 - 10 \\ y = -4 $$ Thus, the point of intersection is \((\frac{3}{2}, -4)\), which is our solution.

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations that we deal with all at once. The ultimate goal is to find a common solution that satisfies these equations simultaneously. In our exercise, we have two linear equations:

• \( y = 4x - 10 \)
• \( y = -2x - 1 \)

These equations form a system because they need to be solved together.

Solving systems of equations can be done using various methods, including substitution, elimination, or graphically. Each method has its own advantages depending on the problem. In this scenario, we are using the graphical method. This involves plotting the lines represented by each equation and finding their point of intersection. The point where they meet is the solution to the system, as it satisfies both equations. This graphical approach provides a visual understanding of how the equations relate to each other.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) represents the y-intercept.

Understanding slope and y-intercept is crucial for graphing lines easily.

In our exercise, the slope-intercept form allows us to identify and plot key components of each line:

• For \( y = 4x - 10 \), the slope \( m \) is \( 4 \), and the y-intercept \( b \) is \( -10 \).
• For \( y = -2x - 1 \), the slope \( m \) is \( -2 \), and the y-intercept \( b \) is \( -1 \).

The slope tells us how steep the line is and in which direction it slants, while the y-intercept shows where the line crosses the y-axis.
This form simplifies the graphing process and ensures an accurate visualization of the system of equations.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph lines and functions to visually solve mathematical problems. It's divided by the x-axis (horizontal) and y-axis (vertical) and is composed of four quadrants.

• The points are denoted as \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position.

Using the coordinate plane, we can easily represent equations and analyze intersections.

For example, in solving our system graphically:

• We start by plotting the y-intercepts of each line: \(-10\) for the first line and \(-1\) for the second line.
• From these intercepts, the slopes give further direction on how the line extends across the plane.

The graphical method's utility lies in its ability to provide an intuitive understanding of where two equations hold simultaneously true by identifying their intersection point on this plane. The intersecting point \((\frac{3}{2}, -4)\) is thus the solution to the system, showing the exact values for \(x\) and \(y\) where both equations are satisfied.