Question

Use linear combinations to solve the system. $$ \begin{aligned} &6 x-5 y=3\\\ &-12 x+8 y=5 \end{aligned} $$

Short answer

The solution is \( x = -\frac{49}{12} \), \( y = -\frac{11}{2} \)

Step-by-step solution

Multiply second equation to make the coefficients of x the same

We begin by making the coefficients of x in both equations to be the same. The easiest way to do this is to multiply the second equation by \( \frac{1}{2} \) to get \( -6x +4y = \frac{5}{2} \)

Add the two equations

We then add the two equations: \( (6x - 5y) + (-6x + 4y) = 3 + \frac{5}{2} \)\. This simplifies to \( -y = \frac{11}{2} \) or \( y = - \frac{11}{2} \)

Substitute y in the first equation

We substitute \( y = -\frac{11}{2} \) into the first equation: \( 6x -5(-\frac{11}{2}) = 3 \) which simplifies to \( 6x +\frac{55}{2} = 3 \) or \( x = -\frac{49}{12} \)

Key concepts

Linear Combinations

To solve systems of linear equations, the method of linear combinations is an efficient technique. This approach involves altering the given equations in such a way that adding or subtracting the modified equations will eliminate one variable. This is possible because in a linear equation, the relationship between the variables is proportional.

Let's consider an example from the textbook. The system consists of two equations:
\[ 6x - 5y = 3 \]
\[ -12x + 8y = 5 \]
Our goal is to eliminate one variable so we can solve for the other. In this case, we choose to eliminate the variable x. To do so, we need the coefficients of x to be opposites; that is, when one equation is multiplied by a certain number, the x-terms will cancel out when the equations are either added to or subtracted from each other.

Once you have achieved this, add the equations together. In the example, multiplying the second equation by \( \frac{1}{2} \) gives us equations with matching coefficients for x (but with opposite signs), leading to their cancellation upon addition and making it easier to solve for y.

Equation Coefficients

In linear equations, the numbers multiplying the variables are known as coefficients. These coefficients determine the slope of the line in a graph and the proportionality between the variables. In the system of equations provided in the textbook example:
\[ 6x - 5y = 3 \]
\[ -12x + 8y = 5 \]
the coefficients are 6 and -5 for the first equation, and -12 and 8 for the second equation. It's crucial to alter these coefficients to create a scenario where one variable can be eliminated when the equations are combined. Here's the kicker: You don't necessarily have to match the coefficients exactly; they just need to be opposites for cancellation. For instance, in the given example, when we multiply the second equation by \( \frac{1}{2} \) it results in the coefficients for x in both equations being -6 and 6, respectively, which are then set to cancel each other out when the equations are added.

Substitution Method

The substitution method is another fundamental technique for solving systems of linear equations. Once you have used linear combinations to eliminate one variable and solve for the other, this first solution must be used to find the value of the second variable.

Continuing with our example above, we first found that \( y = -\frac{11}{2} \). We then substitute this value back into one of the original equations, most commonly the simplest one. In this case, the first equation \( 6x - 5y = 3 \) becomes our choice. Substituting the value of y, we get:
\[ 6x - 5(-\frac{11}{2}) = 3 \]
This simplifies to solve for x. It is crucial to perform substitution carefully to avoid mistakes with signs and operations. If done correctly, you will arrive at the second solution which, together with the first, represents the point where the two lines intersect in a graph - if a unique solution exists.