Solving Linear Equations By Graphing: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of linear equations and how to solve them by graphing. It might sound intimidating, but trust me, it's totally doable. We'll break it down step-by-step, so you can tackle these problems with confidence. We'll specifically look at the system:

$$\begin{array}{l} y + 2.3 = 0.45x \\ -2y = 4.2x - 7.8 \end{array}$$

And determine which of the following options is correct:

A. (2.4, -1.2) B. (-1, 2.5) C. no solution D. infinitely many solutions

Understanding Linear Equations

Before we jump into solving, let's quickly recap what linear equations are. A linear equation is an equation that, when graphed, forms a straight line. The general form of a linear equation is y = mx + b, where m represents the slope of the line, and b represents the y-intercept (the point where the line crosses the y-axis). Understanding this form is crucial because it allows us to easily visualize and manipulate these equations. When we have two or more linear equations, we call it a system of linear equations. The solution to a system of linear equations is the point (or points) where the lines intersect. This point satisfies all equations in the system. If the lines never intersect, there is no solution. If the lines are the same, there are infinitely many solutions. Graphing is a visual method to find these solutions, and it helps in understanding the relationship between the equations.

Step 1: Rewrite the Equations in Slope-Intercept Form

To make graphing easier, we need to rewrite both equations in the slope-intercept form (y = mx + b). This form allows us to quickly identify the slope and y-intercept, which are essential for graphing. Let's start with the first equation:

y + 2.3 = 0.45x

To isolate y, we subtract 2.3 from both sides:

y = 0.45x - 2.3

Now, let's rewrite the second equation:

-2y = 4.2x - 7.8

To isolate y, we divide both sides by -2:

y = -2.1x + 3.9

So now we have our two equations in slope-intercept form:

$$\begin{array}{l} y = 0.45x - 2.3 \\ y = -2.1x + 3.9 \end{array}$$

This form makes it much easier to graph these lines.

Step 2: Graph the Equations

Now that we have our equations in slope-intercept form, we can graph them. You can use graph paper, a graphing calculator, or an online graphing tool. For the first equation, y = 0.45x - 2.3, the y-intercept is -2.3, and the slope is 0.45. This means for every 1 unit we move to the right on the x-axis, we move 0.45 units up on the y-axis. Plotting a couple of points using this information and drawing a line through them gives us the graph of the first equation. Similarly, for the second equation, y = -2.1x + 3.9, the y-intercept is 3.9, and the slope is -2.1. This means for every 1 unit we move to the right on the x-axis, we move 2.1 units down on the y-axis. Plotting a couple of points using this information and drawing a line through them gives us the graph of the second equation. Accurately graphing these lines is essential for finding the correct solution. The more precise your graph, the more accurate your solution will be.

Step 3: Find the Point of Intersection

The solution to the system of equations is the point where the two lines intersect. By looking at the graph, we can estimate the coordinates of this point. It's super important to be as precise as possible when reading the coordinates from the graph. If you're using a graphing calculator or software, you can often find the intersection point directly using built-in functions. However, since we need to round to the nearest tenth, we might not get the exact answer directly from the graph. Eyeballing it, the intersection appears to be around (2.4, -1.2). However, visual estimations can be misleading, so let's verify this algebraically.

Step 4: Verify the Solution Algebraically

To make sure our graphical solution is correct, we can plug the coordinates (2.4, -1.2) into both equations and see if they hold true. This step is crucial because it confirms whether our estimated solution is accurate. Let's start with the first equation:

y + 2.3 = 0.45x

Substitute x = 2.4 and y = -1.2:

-1.2 + 2.3 = 0.45 * 2.4

1.1 = 1.08

This is close, but not exact due to rounding. Now, let's check the second equation:

-2y = 4.2x - 7.8

Substitute x = 2.4 and y = -1.2:

-2 * (-1.2) = 4.2 * 2.4 - 7.8

2.4 = 10.08 - 7.8

2.4 = 2.28

Again, this is close but not exact. The small discrepancies are due to the rounding we did earlier. However, since (2.4, -1.2) is the closest solution when rounded to the nearest tenth, it's the most likely answer.

Step 5: Analyze the Options

Now let's consider the other options. If the lines were parallel, there would be no solution. This would mean the lines never intersect. If the lines were the same, there would be infinitely many solutions. This would mean the lines overlap completely. Since we found a point of intersection (approximately), we can rule out these options.

Final Answer

Based on our graphical analysis and algebraic verification, the solution to the system of linear equations, rounded to the nearest tenth, is:

(A) (2.4, -1.2)

So there you have it! Solving systems of linear equations by graphing can be straightforward once you get the hang of it. Remember to rewrite the equations in slope-intercept form, graph them accurately, find the point of intersection, and verify your solution. Keep practicing, and you'll become a pro in no time! If you have any questions, feel free to ask. Keep up the great work, guys!