True Statements About The System Of Equations: Find 3 Correct Answers

Let's dive into this system of equations problem! We're given a system of two linear equations and need to figure out which three statements about the system are true. It sounds like a fun challenge, so let's get started!

Understanding the System of Equations

First, let's take a good look at the system:

2y = x + 10
3y = 3x + 15

To really understand what's going on, we need to get these equations into a more familiar form. The slope-intercept form (y = mx + b) is super helpful because it lets us easily see the slope (m) and y-intercept (b) of each line. So, that's our first step – let's rewrite these equations!

Converting to Slope-Intercept Form

For the first equation, 2y = x + 10, we need to isolate y. We can do this by dividing both sides of the equation by 2:

2y / 2 = (x + 10) / 2
y = (1/2)x + 5

Okay, equation one is now in slope-intercept form! We can see that the slope of this line is 1/2, and the y-intercept is 5.

Now, let's do the same thing for the second equation, 3y = 3x + 15. We'll divide both sides by 3 to isolate y:

3y / 3 = (3x + 15) / 3
y = x + 5

Awesome, we've got the second equation in slope-intercept form too! Here, the slope is 1 (which is the same as 1/1), and the y-intercept is 5.

Now that we have both equations in slope-intercept form, let's write them together so we can easily compare them:

y = (1/2)x + 5
y = x + 5

Analyzing the Equations

Now comes the fun part – analyzing these equations! What can we tell just by looking at them? The key things to pay attention to are the slopes and the y-intercepts.

• Slopes: The slopes tell us how steep the lines are. If the slopes are the same, the lines are parallel. If the slopes are different, the lines will intersect at some point.
• Y-intercepts: The y-intercepts tell us where the lines cross the y-axis. If the y-intercepts are the same, the lines cross the y-axis at the same point. If the slopes are the same and the y-intercepts are the same, then the lines are actually the same line!

In our case, the slopes are 1/2 and 1, which are different. This means the lines are not parallel; they will intersect. Also, the y-intercepts are both 5, meaning they cross the y-axis at the same spot. This is great information! It tells us a lot about the solution to the system.

Evaluating the Statements

Now that we've analyzed the system, let's think about the statements we're given and see which ones are true. Let's consider some possible statements:

A. The system has one solution. B. The system graphs parallel lines. C. The solution is the intersection of the two lines.

Let's break down each statement based on our analysis:

Statement A: The system has one solution.

Is this true? Well, we determined that the lines have different slopes. Lines with different slopes will always intersect at one point. This point of intersection represents the single solution to the system of equations. So, Statement A is true!

Statement B: The system graphs parallel lines.

Is this true? Parallel lines have the same slope but different y-intercepts. Our lines have different slopes (1/2 and 1), so they are not parallel. Statement B is false.

Statement C: The solution is the intersection of the two lines.

Is this true? Absolutely! The solution to a system of linear equations is the point where the lines intersect. That's where the x and y values satisfy both equations simultaneously. Statement C is true!

Finding the Third True Statement

We've already identified two true statements, but we need to find a third. To do this effectively, we need to find the actual solution to the system. Remember, the solution is the point (x, y) where the two lines intersect. There are a couple of ways we can find this:

1. Substitution: Solve one equation for one variable (e.g., solve for y in terms of x) and substitute that expression into the other equation.
2. Elimination: Multiply one or both equations by a constant so that the coefficients of one variable are opposites. Then, add the equations together to eliminate that variable.

Let's use the substitution method since we already have both equations solved for y:

y = (1/2)x + 5
y = x + 5

Since both equations are equal to y, we can set them equal to each other:

(1/2)x + 5 = x + 5

Now, let's solve for x. First, subtract 5 from both sides:

(1/2)x = x

Next, subtract (1/2)x from both sides:

0 = (1/2)x

Finally, multiply both sides by 2:

0 = x

So, x = 0! Now that we have the x-value, we can plug it into either equation to find the y-value. Let's use the second equation, y = x + 5:

y = 0 + 5
y = 5

Therefore, the solution to the system is the point (0, 5).

Constructing a Third Statement

Now that we have the solution (0, 5), we can create a statement about it. Here's one possible statement:

D. The solution to the system is (0, 5).

This statement is true because we just calculated the solution and found it to be (0, 5).

Conclusion: Three True Statements

Alright, guys, we did it! We've analyzed the system of equations, evaluated the statements, and found three correct answers. The three true statements about the system are:

A. The system has one solution. C. The solution is the intersection of the two lines. D. The solution to the system is (0, 5).

By converting the equations to slope-intercept form, analyzing the slopes and y-intercepts, and solving for the point of intersection, we were able to confidently identify the correct statements. Great job working through this problem! Understanding systems of equations is a crucial skill in algebra, and you've taken a big step forward today. Keep practicing, and you'll become a pro in no time!