No Unique Solution: Finding Equation Pairs Explained

Hey guys! Let's dive into the world of linear equations and figure out which pairs don't have just one single solution. It's like detective work, but with math! We'll break down each pair of equations, so you can totally understand why some have one solution, while others have infinite solutions or no solutions at all. Understanding the conditions for systems of equations is crucial in mathematics. Systems of equations can have one solution, infinitely many solutions, or no solution, depending on the relationship between the equations.

Understanding Linear Equations

First, let's refresh our memory on what linear equations are. Linear equations are equations that can be written in the form y = mx + c, where:

• y is the dependent variable
• x is the independent variable
• m is the slope (or gradient)
• c is the y-intercept (the point where the line crosses the y-axis)

When we have a pair of 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. Graphically, this is where the two lines cross each other on a coordinate plane. But sometimes, things get a little trickier. The number of solutions depends on the relationship between the two lines. They can intersect at one point (one solution), be the same line (infinite solutions), or never intersect (no solution). We'll explore each of these scenarios as we analyze the given pairs of equations. Thinking about slopes and intercepts can help us predict the type of solution we'll find even before we start solving the equations. This way, we can approach the problem more strategically and efficiently.

Conditions for Solutions

There are three possible scenarios when dealing with a pair of linear equations:

1. One Unique Solution: The lines intersect at exactly one point. This happens when the slopes (m) of the two lines are different. Different slopes mean the lines will eventually cross. We love this scenario because we get a definite answer for both x and y! This is the most straightforward case, where the lines have different slopes and intersect at a single, unique point. Imagine two roads crossing each other; they meet at just one intersection.
2. Infinite Solutions: The lines are the same. This means they have the same slope and the same y-intercept. Essentially, it's the same line written in a possibly different form. Any point on one line is also on the other, leading to infinite solutions. These are equations that might look different but are actually the same line in disguise. They have the same slope and y-intercept, meaning they overlap completely. Think of it as two paths drawn perfectly on top of each other; you can't tell them apart.
3. No Solution: The lines are parallel. This means they have the same slope but different y-intercepts. They'll never intersect, so there's no solution that satisfies both equations. This is when the lines have the same slope but different y-intercepts, meaning they run side by side without ever meeting. It's like two parallel train tracks; they never intersect, so there's no common solution.

Now that we've got the basics down, let's apply these concepts to the specific pairs of equations given in the problem.

Analyzing the Equation Pairs

Let's break down each pair of equations to see how many solutions they have.

(1) $2x - 3y = 1$ and $6x - 9y = 3$

First, let’s get these equations into the y = mx + c form. It'll make it easier to compare their slopes and y-intercepts.

• Equation 1: $2x - 3y = 1$
  • Subtract 2x from both sides: $-3y = -2x + 1$
  • Divide by -3: $y = (2/3)x - 1/3$
• Equation 2: $6x - 9y = 3$
  • Subtract 6x from both sides: $-9y = -6x + 3$
  • Divide by -9: $y = (2/3)x - 1/3$

Notice anything? The equations are identical! They have the same slope (2/3) and the same y-intercept (-1/3). This means they represent the same line. So, how many solutions do they have? You guessed it – infinitely many!

(2) $y = 4x$ and $y = -4x$

These equations are already in the y = mx + c form, which is super convenient. Let’s take a look at their slopes and y-intercepts.

• Equation 1: $y = 4x$ (slope = 4, y-intercept = 0)
• Equation 2: $y = -4x$ (slope = -4, y-intercept = 0)

The slopes are different (4 and -4), which means these lines will intersect at one point. Different slopes? That's our key indicator for a unique solution! In this case, they intersect at the origin (0, 0). So, this pair has one unique solution.

(3) $y = 4x$ and $2y - 8x = 8$

Again, let’s get the second equation into y = mx + c form.

• Equation 1: $y = 4x$ (slope = 4, y-intercept = 0)
• Equation 2: $2y - 8x = 8$
  • Add 8x to both sides: $2y = 8x + 8$
  • Divide by 2: $y = 4x + 4$

Now, let's compare. The slopes are the same (4), but the y-intercepts are different (0 and 4). What does that tell us? These lines are parallel! Parallel lines never intersect, so this pair has no solution. No intersection means no solution – simple as that!

(4) $2x - 3y = 1$ and $y = x + 2$

Let’s rewrite the first equation in y = mx + c form, just like we did before.

• Equation 1: $2x - 3y = 1$
  • Subtract 2x from both sides: $-3y = -2x + 1$
  • Divide by -3: $y = (2/3)x - 1/3$
• Equation 2: $y = x + 2$ (slope = 1, y-intercept = 2)

The slopes are different (2/3 and 1). Different slopes mean these lines will intersect at a single point. So, this pair has one unique solution. We're on a roll!

Identifying Pairs with No Unique Solution

Okay, we've analyzed each pair. Now, let's figure out which pairs do not have a unique solution. Remember, that means either infinite solutions (same line) or no solution (parallel lines).

• Pair (1): Infinite solutions (same line)
• Pair (2): One unique solution
• Pair (3): No solution (parallel lines)
• Pair (4): One unique solution

So, pairs (1) and (3) do not have a unique solution. They either have infinite solutions or no solution.

Final Answer

Based on our analysis, the equation pairs that do not have one unique solution are (1) and (3). Therefore, the correct answer is B. (1) and (3).

Isn't it cool how we can predict the number of solutions just by looking at the slopes and y-intercepts? Keep practicing, and you'll become a pro at solving systems of equations in no time!