Question

Solving with Elimination Quick Check
5 of 55 of 5 Items








Question
How many solutions does the system have?
12x−15y=18

4x−5y=6
(1 point)
Responses

one solution
one solution

infinitely many solutions
infinitely many solutions

two solutions
two solutions

no solution

Answer 1

To determine how many solutions the system of equations has, we can first analyze the given equations:

1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)

We'll use the elimination method to find out if the equations are consistent and how many solutions they have.

### Step 1: Simplify the equations if necessary
Let's simplify the second equation:

The equation \( 4x - 5y = 6 \) can be multiplied by 3 to align it with the coefficients in the first equation:

\[
3(4x - 5y) = 3(6)
\]

This gives us:

\[
12x - 15y = 18
\]

### Step 2: Compare the equations
Now we have:

1. \( 12x - 15y = 18 \)
2. \( 12x - 15y = 18 \)

Since both equations are identical, all points that satisfy one equation will satisfy the other. Hence, there are infinitely many solutions.

### Conclusion
The system of equations has **infinitely many solutions**.