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Math  /  Algebra

QuestionWhich linear equation has more solutions? One point. 3) y=x−4y=x-4 or y=10x−40y=10 x-40

Studdy Solution

STEP 1

What is this asking? Which of these two lines has more points on it? Watch out! It's easy to think that (\$y = 10x - 40\$ looks longer because of the bigger number, but that's not what we're looking for!

STEP 2

1. (Understand Linear Equations
2. (Compare the Lines

STEP 3

Let's remember what a linear equation *is*! It's a relationship between \(x and yy that makes a **straight line** when you graph it.
Any point on that line is a **solution** to the equation.

STEP 4

How many points are on a line?
Infinitely many!
Lines go on forever in both directions, so there are endless points we could find.

STEP 5

Both of our equations, y=x−4y = x - 4 and y=10x−40y = 10x - 40, are **linear equations**.
This means they both represent straight lines, and both have an **infinite number** of solutions.

STEP 6

Let's look at y=x−4y = x - 4.
If we plug in x=0x = 0, we get y=0−4=−4y = 0 - 4 = -4.
So, (0,−4)(0, -4) is one point on the line.
If we plug in x=1x = 1, we get y=1−4=−3y = 1 - 4 = -3.
So, (1,−3)(1, -3) is another point!
We could do this forever!

STEP 7

Now let's look at y=10x−40y = 10x - 40.
If we plug in x=0x = 0, we get y=10⋅0−40=−40y = 10 \cdot 0 - 40 = -40.
So, (0,−40)(0, -40) is one point on *this* line.
If we plug in x=1x = 1, we get y=10⋅1−40=−30y = 10 \cdot 1 - 40 = -30.
So, (1,−30)(1, -30) is another point!
Again, we could do this forever!

STEP 8

Since both lines go on forever, they both have infinitely many points, and therefore, infinitely many solutions!

STEP 9

Both equations have the **same number** of solutions: infinitely many!

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