Question

Solve the system of equations.
x = 2y-5
-3x = -6y+15

Answer 1

Infinitely Many Solutions

Explanation:

We are given the system of equations:

`\displaystyle \large{\begin{cases} x = 2y-5 \\ -3x=-6y+15 \end{cases}}`

Since the first equation is already in x-isolated form, we can use the first equation to substitute in the second equation.

Substitute x = 2y-5 in the second equation.

`\displaystyle \large{-3(2y-5)=-6y+15}`

Then we solve the equation for y-value; distribute -3 in.

`\displaystyle \large{-6y + 15=-6y+15}`

We have to combine like terms. First, add both sides by 6y.

`\displaystyle \large{-6y + 6y+ 15=-6y + 6y+15} \\ \displaystyle \large{ 15=15}`

Looks like both sides are equal. If both sides are equal and no variables exist, the equation is always true for all real numbers.

`\displaystyle \large{y \in \R}`

Since y is true for all real numbers. That means x is also true for all real numbers. That's because for x = 2y-5, we can substitute y = any real numbers and still get the x-value. Therefore:

`\displaystyle \large{x \in \R}`

Since it's the system of equations where we find the intercept(s) of graphs, the solutions for x and y are all real numbers, that means both graphs are same and intersect each others infinitely.

Let's take a look at both equations again this time.

`\displaystyle \large{\begin{cases} x = 2y-5 \\ -3x=-6y+15 \end{cases}}`

For the second equation, we can divide the whole equation by -3.

`\displaystyle \large{\begin{cases} x = 2y-5 \\ ( - 3x)/( - 3) = ( - 6y)/( - 3) + (15)/( - 3) \end{cases}} \\ \displaystyle \large{\begin{cases} x = 2y-5 \\ x = 2y - 5 \end{cases}}`

Notice that two equations are same and therefore, both graphs are same and intersect each others infinitely.

Hence, the solution to this answer is Infinitely Many Solutions