Question

Decide which variable to eliminate when using linear combinations to solve the system. Explain your thinking. $$\begin{aligned} &2 x+3 y=1\\\ &4 x-2 y=10 \end{aligned}$$

Short answer

The variable to eliminate is x. When using linear combinations to solve the system, it is more straightforward to eliminate the x variable, resulted y = -1.

Step-by-step solution

Multiply the first equation by 2

In order to eliminate variable x, the first equation needs to be multiplied by 2, this resuls in 4x + 6y = 2

Subtract the second equation from the new equation from Step 1

Subtract the second equation from the new equation made in step 1. The operation will be (4x + 6y = 2) - (4x - 2y = 10). After performing the subtraction, the result is 8y = -8

Solve for y

Having an equation with only one unknow variable (y), we can easily solve for y. This is done by dividing both sides of the equation by 8, the result is y = -1

Key concepts

Linear Combination

Linear combination is a fundamental technique in algebra that allows us to add or subtract equations in a system to eliminate variables. It's a bit like strategically stacking blocks so that similar ones cancel each other out. In our exercise, we have two equations with two variables, x and y, which we can think of as two different types of blocks. The goal is to manipulate these 'blocks' (equations) to simplify the complex structure (system of equations) into something more straightforward.
By carefully choosing coefficients to multiply each equation by, we create an opportunity to cancel one variable when we add or subtract the equations from one another. In the given problem, we see this in action when we decide which variable to eliminate. The choice depends on the simplicity of the calculations and the potential for reducing the system to a single variable equation. By multiplying the first equation by 2, we align the coefficients of x in both equations, thus preparing the path to eliminate x through subtraction. This process transforms the complex system into a simpler one that is much easier to solve.

Variable Elimination

Variable elimination is the heart of solving systems of equations by elimination. This strategy revolves around removing one variable to focus on the other, much like removing puzzle pieces to see a clearer picture of what's missing. In our textual puzzle of equations, eliminating a variable reduces the mental load required to see the solution.
To eliminate a variable, we need to create a zero coefficient for that variable in one of the equations. This is done by making the coefficients of the chosen variable exact opposites in two equations, so when we perform addition or subtraction, they cancel out. In our example, the multiplication of the first equation by 2 sets the stage for the x variable to be eliminated. When we subtract the second equation from the first, x vanishes like magic, because the coefficients, both now 4, negate each other. Variable elimination is a powerful trick in algebra's bag that simplifies the problem into a single-variable equation, providing us with a direct route to the solution.

Algebraic Manipulation

Algebraic manipulation encompasses the various techniques used to rearrange, simplify, or rewrite expressions and equations. It is the skill of an algebraic chef—mixing ingredients to perfect the recipe. In the realm of systems of equations, it involves operations like addition, subtraction, multiplication, and division, carefully applied to modify and solve the equations.
In our exercise, after eliminating x, we are left with an equation that features only y. This single-variable equation is straightforward to solve using basic algebraic manipulation. Dividing both sides of the equation by the coefficient of y isolates this variable, revealing its value. Here, the straightforward division of -8 by 8 yields y = -1, a satisfying end to our algebraic journey. Mastering algebraic manipulation enables students to approach complex problems with confidence, knowing they have the tools to simplify and solve.