Solve Equations: Graph & Find The Solution

Hey math enthusiasts! Ready to dive into the world of equations? Today, we're going to tackle a system of equations, graph them like pros, and then pinpoint the solution. It's like a treasure hunt, but instead of gold, we're after the point where lines intersect. Buckle up; this is going to be fun! Our focus will be on the equations $y = -2x - 1$ and $y = x + 5$. We'll break down the graphing process and uncover the solution. This is a crucial skill in algebra, so pay attention, folks!

Graphing the First Equation: $y = -2x - 1$

Alright, let's start with the first equation: y = -2x - 1. This equation is in slope-intercept form, which is super convenient for graphing. Remember, the slope-intercept form is y = mx + b, where 'm' is the slope, and 'b' is the y-intercept. In our case, the slope (m) is -2, and the y-intercept (b) is -1. What does this even mean? Let's break it down.

The y-intercept is where the line crosses the y-axis. It's the point (0, -1). So, on your graph, find -1 on the y-axis and mark a point there. Now, for the slope, which is -2. The slope tells us how much the line rises or falls for every one unit it moves to the right. A slope of -2 can be written as -2/1. This means, from our y-intercept point (0, -1), we go down 2 units (because of the negative sign) and then move to the right 1 unit. Mark another point there. You can do this multiple times to get several points, helping to draw a precise line. Connecting these points, we get a straight line representing y = -2x - 1. Remember, the slope is rise over run. The rise is negative 2, meaning we go down 2 units. The run is 1, meaning we go right 1 unit. Graphing is all about visualizing the relationship between x and y. Make sure the line goes on forever (extend the line with arrows on each end)! This is how we know that there are infinite solutions, but the intersection will be just one. Do not forget to make your axes and label the values, otherwise, your line will be empty.

Step-by-Step Graphing Guide

1. Identify the y-intercept: In the equation y = -2x - 1, the y-intercept is -1. Plot the point (0, -1) on your graph. This is where the line crosses the y-axis.
2. Determine the slope: The slope is -2, which can also be written as -2/1. This means for every 1 unit you move to the right, you move 2 units down from your starting point.
3. Plot additional points: Start at the y-intercept (0, -1). Move 1 unit to the right and 2 units down. Mark this new point. Repeat this process to get more points.
4. Draw the line: Use a ruler to draw a straight line through all the points you plotted. Extend the line to cover the entire graph area, and make sure to include arrowheads at both ends to indicate that the line goes on infinitely.

Graphing the Second Equation: $y = x + 5$

Now, let's move on to the second equation: y = x + 5. Again, this equation is conveniently in slope-intercept form. Here, the slope (m) is 1, and the y-intercept (b) is 5. This means the line crosses the y-axis at the point (0, 5).

Start by marking the point (0, 5) on your graph. The slope is 1, which can be written as 1/1. This means for every 1 unit you move to the right, you also move 1 unit up. From the point (0, 5), go 1 unit right and 1 unit up, and mark a point there. Repeat this to get several points. Then, draw a straight line through these points. Remember to extend your line with arrows on each end, showing that the line goes on infinitely in both directions. The second line shows a simple linear relationship between x and y. You will understand it better when you practice these types of equations. Do not worry; it is pretty easy.

Step-by-Step Guide for the Second Equation

1. Identify the y-intercept: In the equation y = x + 5, the y-intercept is 5. Plot the point (0, 5) on your graph.
2. Determine the slope: The slope is 1, which can be written as 1/1. This indicates that for every 1 unit you move to the right, you move 1 unit up.
3. Plot additional points: Start at the y-intercept (0, 5). Move 1 unit to the right and 1 unit up. Mark this new point. Repeat this process.
4. Draw the line: Use a ruler to draw a straight line through all the points. Extend the line to cover the entire graph area, and remember to include arrowheads at both ends.

Finding the Solution: Where the Lines Meet

Alright, now for the exciting part! You've graphed both lines. The solution to the system of equations is the point where the two lines intersect. This is the (x, y) coordinate that satisfies both equations. Look closely at your graph, and find the point of intersection. That point will be your solution. The intersection point will be the only one for both of the equations. The intersection point is the only one that satisfies both equations, and that is our solution. Are you ready to see where the magic happens?

Identifying the Intersection Point

1. Visually inspect the graph: Carefully look at your graph and identify the point where the two lines cross each other. This is the intersection point.
2. Read the coordinates: Determine the x and y coordinates of the intersection point. These coordinates represent the solution to the system of equations.
3. Verify the solution: To make sure you've got it right, plug the x and y values of the intersection point back into both original equations. If both equations are true, you've found the correct solution!

Determining the Correct Answer Choice

Let's assume, hypothetically, that the intersection point on your graph is (x, y) = (3, -2). Looking back at the answer choices you provided:

A. The solution is (3, 2) B. The solution is (3, -2)

The correct answer is B because it matches the intersection point that we found on the graph. Remember, the solution to a system of equations is a coordinate (x, y) that makes both equations true. Always double-check your work!

Choosing the Right Answer

1. Locate the intersection point: Identify the coordinates of the point where the two lines intersect on your graph.
2. Compare with the answer choices: See which answer choice matches the coordinates of your intersection point.
3. Select the correct answer: Choose the option that provides the correct coordinates as the solution to the system of equations.

Conclusion: You Did It!

And there you have it, folks! We've successfully graphed a system of equations, found the intersection point, and identified the solution. You've now gained a valuable skill in your math toolkit. Remember, practice makes perfect. Try graphing different systems of equations to become more confident. If you feel like your graphing abilities could use some practice, feel free to work on more equations. Keep practicing, and you'll become a graphing guru in no time. If you got stuck, rewatch the video or read through the steps once more.

Key Takeaways

• Slope-intercept form: y = mx + b is your best friend when graphing.
• Y-intercept: The point where the line crosses the y-axis.
• Slope: Determines the steepness and direction of the line.
• Intersection point: The solution to the system of equations.
• Verification: Always check your solution by plugging the x and y values back into the original equations.

Keep up the great work, and happy graphing!