Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &y-2 x=0\\\ &6 y+2 x=0 \end{aligned}$$

Short answer

The solution to the system of equations is \(x=0\) and \(y=0\).

Step-by-step solution

Rewrite the first equation

First, the first equation is rearranged to isolate y. This gives: \( y = 2x \)

Substitute into second equation

The result from the first step is now substituted into the second equation, replacing y with the expression found. That leads to: \( 6(2x) + 2x = 0 \)

Solve the equation

The equation in step 2 simplifies to \( 12x + 2x = 0 \). Combining like terms yields \( 14x = 0 \). When solved for x, this equation gives \( x = 0 \)

Substitute x into first equation

Now, the value of x found in step 3 is substituted back into the first equation yields: \(y = 2(0)\)

Solve for y

Solving the equation in step 4 for y gives \( y = 0 \)

Key concepts

Linear Combinations

Understanding linear combinations is pivotal when solving a system of linear equations. It refers to the process of adding or subtracting equations in order to eliminate one of the variables, making it easier to solve for the remaining one.

In the provided exercise, we have two equations: the first being y - 2x = 0 and the second 6y + 2x = 0. To use linear combinations, we can add or subtract these equations such that one variable is eliminated. In this case, adding the two equations gets rid of the x variable because -2x and +2x cancel each other out, leading to 7y = 0. As a method, linear combinations involve the strategic manipulation of equations to create a simpler equation that is solvable. This technique is particularly useful when dealing with larger systems of equations or when variables conveniently cancel each other out.

Substitution Method

The substitution method is another fundamental technique used in solving systems of equations. It involves solving one of the equations for one variable and then substituting the result into the other equation.

Looking at step 1 of our example, y = 2x is derived from the first equation. This expression for y is then substituted into the second equation during step 2. This substitution simplifies the problem to a single variable equation, which is easier to solve. Notably, the substitution method is most efficient when one of the equations can be rearranged easily to express one variable in terms of the other, as was the case with our first equation. It's essential to perform substitutions accurately to maintain the equivalence of the original system. The substitution method also demonstrates the interconnectedness of the variables within a system, showing that the value of one directly influences the other.

Combining Like Terms

When solving equations, combining like terms helps simplify expressions, making it easier to find the solution. 'Like terms' are terms that have the exact same variables raised to the same power, though they may have different coefficients.

In step 3, the equation obtained from substitution, 12x + 2x = 0, has like terms which are combined to produce 14x = 0. To combine like terms, you simply add or subtract the coefficients and keep the variable portion unchanged. This necessary step moves you towards a more manageable equation that typically leads to a solution. Being adept at combining like terms is crucial not just for solving linear equations but also in simplifying expressions across various algebraic applications.