Question

What variable do I solve for first? Do I solve for X because it’s first in the alphabet? If so, which X variable do I solve for? Do I solve for the smallest or biggest? I added a photo of the one i’m struggling with! :))

Answer 1

• first you look at what you have, then develop a strategy
• this system has No Solution

Explanation:

You want to know what to do first with the system of equations ...

• 3.25x -1.5y = 1.25
• 13x -6y = 10

Strategies

You are taught a couple of strategies for solving systems of linear equations algebraically. The "substitution" strategy requires you use one of the equations to write an expression that can be substituted into the other equation. The purpose of this is to reduce the number of variables in the remaining equation. Usually, that means solving for one of the variables to obtain the expression to substitute.

Another strategy you are taught is the "elimination" strategy. It is also called the "addition" or "combination" strategy. It is executed by adding (or subtracting) some multiple of one of the equations from the other equation, or some multiple of it. The purpose of this is to make the coefficient of one of the variables be zero in the combined equation.

Look

Substitution

The substitution strategy is easiest to execute if you already have one or both equations in "y=" or "x=" form. It is nearly as easy to execute if the coefficient of one of the variables is +1 or -1, or if that can be easily made to be the case. So, this is what you look for to see if the substitution strategy is an appropriate choice.

Elimination

The elimination strategy is easiest to execute if the coefficients of one of the variables are the same or opposites. If they are the same, that variable can be eliminated by subtracting one equation from the other. If they are opposites, the variable can be eliminated by adding one equation to the other. So, this relation between coefficients is one of the next things you look for when deciding what your strategy will be.

The elimination strategy can also be effectively used if the coefficients of one of the variables are a nice (integer) multiple of one another. In this problem, we notice that the coefficients of y are -1.5 and -6, which are related by a factor of 4. (It is helpful to be very familiar with multiplication facts.) As it happens, the coefficients of x have the same relation: 13 is 4 times 3.25.

Dependent/Inconsistent

The fact that both sets of coefficients are related by the same factor raises a red flag regarding these equations. It means they are either dependent (have infinite solutions) or are inconsistent (have no solution).

The equations are dependent if one equation is a multiple of the other. Here, we can check that by multiplying the first equation by 4:

4 × (3.25x -1.5y) = 4 × 1.25

13x -6y = 5

We notice the other equation is ...

13x -6y = 10

Values of x and y that make 13x-6y=5 cannot also make that same sum be 10. These are called "inconsistent" equations, and they have No Solution.

Hypothetical: If the first equation were 3.25x -1.5y = 2.5, then multiplying it by 4 would give 13x -6y = 10, the same as the second equation. In this case, the equations would be called "dependent," and any of the infinite number of solutions to the first equation would also be a solution to the second equation.

Plan

After you look at the equations to determine if any of the coefficients are 1, or have nice relations with the coefficients of the other equation, you can formulate a strategy for elimination or substitution. As we saw above, it can be useful to eliminate any fractions to start with. Sometimes, it is also useful to factor out any common factors. For example, 2x + 6y = 8 can be reduced to x +3y = 4 by factoring out 2 from every term.

Then, the variable that you solve for first will be the one that is left after you have done your substitution or elimination.

This System

As we saw above, the given equations can be rewritten as ...

• 13x -6y = 5 . . . . . . first equation multiplied by 4
• 13x -6y = 10

We already know the same coefficients and different constants mean these equations are inconsistent and have No Solution. If we need further convincing we can subtract one from the other. Here, too, we can plan ahead a little bit: subtracting the first from the second will leave a positive constant:

(13x -6y) -(13x -6y) = (10) -(5)

0 = 5 . . . . . . . simplify (false)

There are no values of x and y that will make this false statement true, hence no solution.

Answer 2

No solution

Explanation:

Given the simultaenous equations:

`\displaystyle{\begin{cases} 3.25x - 1.5y = 1.25 \\ 13x-6y=10 \end{cases}}`

The first equation can be rewritten as:

`\displaystyle{(325)/(100) x - (15)/(10)y = (125)/(100)}`

Clear the denominators by multiplying both sides with 100:

`\displaystyle{(325)/(100) x \cdot 100 - (15)/(10)y \cdot 100= (125)/(100) \cdot 100}\\\\\displaystyle{325x - 150y = 125}`

The whole terms are multiple of 25, so we can simplify even lower by dividing both sides by 25:

`\displaystyle{(325x)/(25) - (150y)/(25) = (125)/(25)}\\\\\displaystyle{13x-6y=5}`

So now, we have:

`\displaystyle{\begin{cases}13x-6y=5\\ 13x-6y=10 \end{cases}}`

There are various ways to solve simultaenous equations but I'll use the elimination method. Multiply negative in either first or second equation but I'll choose first:

`\displaystyle{\begin{cases}-13x+6y=-5\\ 13x-6y=10 \end{cases}}`

Add both equations with like terms:

`\displaystyle{0=5}`

Since this equation is false. Therefore, there is no solution to the simultaenous equation.