Solve System Of Equations By Graphing
Hey guys! Today, we're going to tackle a classic math problem: solving a system of equations by graphing. It's a super useful skill, and once you get the hang of it, you'll be able to solve these problems in no time. We'll walk through each step, making sure everything is crystal clear. So, let's dive in!
Understanding the Equations
Before we start graphing, let's take a closer look at our equations. We have:
y - 13 = 2xy - 6x = 29
The goal here is to find the values of x and y that satisfy both equations simultaneously. Graphing is a visual way to find this solution – it's where the two lines intersect. First, we need to get both equations into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. This form makes it super easy to plot the lines on a graph. Let's start with the first equation.
For the first equation, y - 13 = 2x, we need to isolate y. To do that, we simply add 13 to both sides of the equation. This gives us y = 2x + 13. Now it’s in slope-intercept form! We can see that the slope (m) is 2 and the y-intercept (b) is 13. This means the line crosses the y-axis at the point (0, 13), and for every one unit we move to the right on the x-axis, the line goes up two units on the y-axis. Understanding the slope and y-intercept is crucial for accurately graphing the line. We'll use this information in just a bit when we start plotting points.
Now, let's transform the second equation, y - 6x = 29, into slope-intercept form. Again, our aim is to isolate y. To do this, we add 6x to both sides of the equation. This results in y = 6x + 29. Great! This equation is also now in the form y = mx + b. Here, the slope (m) is 6, and the y-intercept (b) is 29. This means the line crosses the y-axis at the point (0, 29), which is quite a bit higher than the first line's y-intercept. Also, for every one unit we move to the right on the x-axis, this line goes up six units on the y-axis, making it steeper than the first line. Recognizing these characteristics will help us sketch the lines accurately and predict where they might intersect. With both equations now in slope-intercept form, we are ready to graph them and find the point of intersection, which will give us our solution.
Graphing the Equations
Alright, now that we have both equations in the y = mx + b form, let’s get them graphed. Remember, the first equation is y = 2x + 13, and the second is y = 6x + 29. You can use graph paper, a graphing calculator, or an online graphing tool – whatever works best for you. The key is to plot the lines accurately so we can find where they intersect.
Graphing the First Equation: y = 2x + 13
To graph this, we start with the y-intercept, which is 13. This means we put a point on the y-axis at the value of 13. From there, we use the slope to find another point. The slope is 2, which can be thought of as 2/1. This means for every 1 unit we move to the right on the x-axis, we move 2 units up on the y-axis. So, starting from the y-intercept (0, 13), we move 1 unit to the right and 2 units up, placing our next point at (1, 15). We can repeat this process to get a few more points, such as (2, 17) and (3, 19). Once you have a few points, use a ruler or straight edge to draw a line through them. Extend the line across the graph. Make sure your line is straight and passes through all the points you plotted. Accurate plotting is crucial for finding the correct intersection point later on.
Graphing the Second Equation: y = 6x + 29
Next, we graph the second equation, y = 6x + 29. This one has a y-intercept of 29, which means it starts much higher on the y-axis. Place a point at (0, 29). The slope is 6, or 6/1, meaning for every 1 unit we move to the right on the x-axis, we move 6 units up on the y-axis. Starting from the y-intercept (0, 29), we move 1 unit to the right and 6 units up, placing our next point at (1, 35). You might notice that this line is much steeper than the first one. Because the y-intercept is quite high, and the slope is steep, you might need to adjust your graph scale to fit this line properly. Draw a line through these points, extending it across the graph. Make sure this line is also straight and accurately represents the equation. With both lines now graphed, we can visually identify their intersection point, which will give us the solution to the system of equations.
Finding the Intersection Point
Now that we've graphed both lines, the next step is to find the point where they intersect. This point represents the solution to the system of equations because it's the only point that satisfies both equations simultaneously. Look closely at your graph and identify the coordinates of the intersection point. It might not always be a perfect integer, so do your best to estimate the values accurately.
In our case, the two lines, y = 2x + 13 and y = 6x + 29, intersect at the point (-4, 5). This means that x = -4 and y = 5. To verify this is the correct solution, we can substitute these values back into the original equations and see if they hold true.
Let's check the first equation: y - 13 = 2x. Plugging in our values, we get 5 - 13 = 2(-4), which simplifies to -8 = -8. This is true, so the point satisfies the first equation. Now let's check the second equation: y - 6x = 29. Plugging in our values, we get 5 - 6(-4) = 29, which simplifies to 5 + 24 = 29, and further to 29 = 29. This is also true, confirming that the point satisfies the second equation as well. Since the point (-4, 5) satisfies both equations, it is indeed the solution to the system of equations.
The Solution
Alright, we've made it to the end! After graphing both equations and finding their intersection point, we've determined the solution to the system of equations.
The solution is: (x, y) = (-4, 5)
So, the values that satisfy both equations y - 13 = 2x and y - 6x = 29 are x = -4 and y = 5. Graphing is a fantastic way to visually confirm your solutions and understand how different equations relate to each other. Great job, guys! You've successfully solved the system of equations by graphing!