Solve the system of equations using elimination.

Question

−
2
x
+
6
y
=
−
4
−
4
x
+
12
y
=
−
8
−2x+6y=−4
−4x+12y=−8

What is the LCM of the x values?


What is the LCM of the y values?


(b)
Write your equivalent equations.

First equation:


Second equation:

(c)
What operation did you use to elimnate?
subtraction


(d)
How many solutions does this system have?

Answer 1

To solve the given system of equations using elimination, we have the following two equations:

1. \(-2x + 6y = -4\)
2. \(-4x + 12y = -8\)

(a) Finding the LCM of x values and y values:

First, let's rewrite the equations for clarity:

1. \(-2x + 6y = -4\) (Equation 1)
2. \(-4x + 12y = -8\) (Equation 2)

Now let's see the coefficients of both equations:

• The coefficients of the variable \( x \) are \(-2\) (from the first equation) and \(-4\) (from the second equation).
• The coefficients of the variable \( y \) are \(6\) (from the first equation) and \(12\) (from the second equation).

LCM of the x values: To find the LCM of \(-2\) and \(-4\), we take their absolute values.

• The absolute values are \(2\) and \(4\).
• The LCM of \(2\) and \(4\) is \(4\).

LCM of the y values: To find the LCM of \(6\) and \(12\):

• The LCM of \(6\) and \(12\) is \(12\).

So:

• LCM of the x values: 4
• LCM of the y values: 12

(b) Equivalent equations

Notice that the second equation can be simplified:

Dividing the second equation by \(-4\): \[ -4x + 12y = -8 \implies x - 3y = 2 \]

Both equations can thus be written as follows:

1. First equation (remains the same): \[ -2x + 6y = -4 \]

2. Equivalent form of the second equation: \[ x - 3y = 2 \]

(c) Operation used to eliminate

We can multiply the first equation by \(2\) to align the coefficients of \(x\), allowing us to eliminate \(x\):

Multiplying the first equation by \(2\):

• \(2(-2x + 6y = -4) \implies -4x + 12y = -8\)

Now we can subtract the second equation from this modified first equation. Hence we used subtraction to eliminate.

(d) Number of solutions of this system

Upon simplifying, we see that both original equations represent the same line, as multiplication of the first equation by \(-2\) yields the second equation. Therefore, the system has an infinite number of solutions (all points on the line defined by the equations).

In summary:

• (a) LCM of x values: 4
• LCM of y values: 12
• Equivalent equations:
  • First: \( -2x + 6y = -4 \)
  • Second: \( x - 3y = 2 \)
• Operation to eliminate: subtraction
• Number of solutions: infinite solutions