One Solution System: Find The Other Equation!

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Let's dive into the world of systems of equations, guys! Specifically, we're going to tackle a problem where we need to figure out what the other equation could be if we already know one equation and that the system has only one solution. Sounds like fun, right? Buckle up, because we're going to break it down step by step.

Understanding Systems of Equations

First, let's make sure we're all on the same page about what a system of equations actually is. A system of equations is simply a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for the variables that make all the equations in the system true simultaneously. Think of it like finding the sweet spot where all the equations agree.

Now, when we talk about the number of solutions a system can have, there are three possibilities:

  1. One Solution: This means the lines represented by the equations intersect at exactly one point. This point is the unique solution to the system.
  2. No Solution: This happens when the lines are parallel and never intersect. They have the same slope but different y-intercepts, so they never cross paths.
  3. Infinite Solutions: This occurs when the equations represent the same line. They overlap completely, meaning every point on the line is a solution to both equations.

In this particular problem, we're told that our system has one solution. This is super important information because it tells us that the lines representing our equations must intersect at a single point. They can't be parallel (no solution) and they can't be the same line (infinite solutions).

The Key: Slopes and Intercepts

The secret to solving this kind of problem lies in understanding the relationship between the slopes and y-intercepts of the lines. Remember the slope-intercept form of a linear equation? It's y = mx + b, where m is the slope and b is the y-intercept. This form is our best friend when we're dealing with systems of equations.

For a system to have one solution, the lines must have different slopes. This is the golden rule! If the slopes are different, the lines will intersect somewhere, giving us our one and only solution. If the slopes are the same, we need to check the y-intercepts. If the y-intercepts are also the same, we have the same line (infinite solutions). If the slopes are the same but the y-intercepts are different, we have parallel lines (no solution).

Analyzing the Given Equation

Okay, let's get down to business. We're given one equation: 4x - y = 5. To make our lives easier, let's rewrite this equation in slope-intercept form (y = mx + b). We can do this by isolating y:

  1. Subtract 4x from both sides: -y = -4x + 5
  2. Multiply both sides by -1: y = 4x - 5

Now we can clearly see that the slope of this line is 4 and the y-intercept is -5. This is crucial information! We need to find another equation that, when paired with this one, will give us a system with exactly one solution. Remember, this means the other equation must have a different slope than 4.

Evaluating the Answer Choices

Now, let's take a look at the answer choices and see which one fits the bill. We're looking for an equation that, when written in slope-intercept form, has a slope that is not equal to 4.

Let's go through the options one by one:

A. 2y = 8x - 10 Divide both sides by 2: y = 4x - 5 The slope is 4. This is the same line as our original equation, so it has infinite solutions. Nope!

B. -2y = -8x - 10 Divide both sides by -2: y = 4x + 5 The slope is 4. Same slope, different y-intercept. Parallel lines, no solution. Not what we want!

C. y = -4x + 5 The slope is -4. Ding ding ding! This is different from 4. This equation will give us a system with one solution!

D. y = 4x - 5 The slope is 4. Same as our original equation. Infinite solutions again. Not the answer.

The Solution

Based on our analysis, the correct answer is C. y = -4x + 5. This equation has a slope of -4, which is different from the slope of our given equation (which is 4). Therefore, these two equations will intersect at one point, giving us a system with exactly one solution.

Key Takeaways

So, what did we learn today, guys? Here's the lowdown:

  • A system of equations with one solution means the lines intersect at one point.
  • For a system to have one solution, the equations must have different slopes.
  • Rewrite equations in slope-intercept form (y = mx + b) to easily identify the slope and y-intercept.
  • Carefully compare the slopes of the equations to determine the number of solutions.

By understanding these concepts, you'll be able to confidently tackle any system of equations problem that comes your way! Keep practicing, and you'll become a system-solving superstar!

Practice Problems

To solidify your understanding, try these practice problems:

  1. One equation in a system is y = 2x + 3. Which of the following could be the other equation if the system has one solution?

    • A. y = 2x - 1
    • B. y = -2x + 3
    • C. 2y = 4x + 6
    • D. y = 2x + 3

  2. One equation in a system is x + y = 4. Which of the following equations would result in a system with no solution?

    • A. 2x + 2y = 8
    • B. x - y = 4
    • C. 2x + 2y = 10
    • D. -x - y = -4

Wrapping Up

Solving systems of equations might seem tricky at first, but with a little practice and a solid understanding of the concepts, you'll be a pro in no time! Remember to focus on the slopes and y-intercepts, and you'll be able to determine the number of solutions with ease. Keep up the great work, and I'll catch you in the next math adventure!