Problem 60 - Algebra 2

Question

Solve each system by elimination or substitution. $$ \left\\{\begin{aligned} 2 x+4 y &=-8 \\\\-5 x+4 y &=6 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution to the system is x = -2, y = -1.

Step 1: Choose a Method - Elimination

We choose the elimination method because the coefficients of y are the same in both equations. This makes it convenient to directly eliminate the y variable by subtracting one equation from the other.

Step 2: Subtract the Equations

Subtract the second equation from the first one to eliminate the y variable. (2x + 4y) - (-5x + 4y) = -8 - 6. This simplifies to 2x + 5x = -14.

Step 3: Solve for x

Combine like terms to solve for x. 2x + 5x = 7x, which gives us the simplified equation 7x = -14. Dividing both sides by 7 yields x = -2.

Step 4: Substitute x value into one of the original equations

Now that we have the value of x, substitute it back into one of the original equations to solve for y. Using the first equation: 2(-2) + 4y = -8.

Step 5: Solve for y

Simplify the equation and solve for y. -4 + 4y = -8, which simplifies to 4y = -4 after adding 4 to both sides. Dividing both sides by 4 yields y = -1.

Key Concepts

Elimination Method

The elimination method is a strategy used to solve systems of linear equations. It involves combining the equations in a way that allows one variable to be eliminated. This is done by ensuring that the coefficients of one of the variables are opposites or the same, which then allows you to add or subtract the equations from one another.

For example, consider the equations given: $$\left\{\begin{aligned} 2x + 4y &= -8 \ -5x + 4y &= 6 \end{aligned}\right.$$
In this case, the coefficients of y are the same in both equations, so when you subtract one equation from the other, the y variable is eliminated. This simplification results in an equation with only one variable, x, which can then be solved easily. The elimination method is particularly useful when the equations are set up in such a way that cancellation of a variable is straightforward.

Substitution Method

The substitution method is another technique for solving systems of linear equations. This method involves solving one of the equations for one variable and then 'substituting' this expression into the other equation. Essentially, you are reducing the system of equations to a single equation with one variable.

Using the example from the elimination exercise, if we instead used substitution, we could solve one of the equations for y and then replace y in the other equation with this expression. For instance: $$2x + 4y = -8$$isolating for y yields:$$4y = -8 - 2x$$Dividing by 4 gives:$$y = -2 - \frac{1}{2}x$$Then, we would 'substitute' this expression for y into the other equation and solve for x. Substitution is particularly advantageous when the coefficients of variables are not well-suited for elimination or when the equations are not conveniently arranged for elimination.

Algebraic Equations

Algebraic equations are mathematical statements that use variables, like x and y, and constants, such as numbers, to show a relationship between quantities. The goal of solving an algebraic equation is to find the value(s) for the variable(s) that make the equation true.

They can range from simple linear equations with one variable to more complex equations involving multiple variables and higher powers. For instance, the system of equations from our exercise is a set of linear algebraic equations since the highest power of the variables x and y is 1. To solve these equations, we use methods like elimination or substitution to systematically isolate and solve for the variables.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal when solving such a system is to find values for the variables that satisfy all equations simultaneously.

For example, the system $$\left\{\begin{aligned} 2x + 4y &= -8 \ -5x + 4y &= 6 \end{aligned}\right.$$is a system of linear equations. Solutions may be a single ordered pair, multiple pairs, or none at all, depending on whether the lines intersect at one point, are the same line, or are parallel lines, respectively. Solving a system of linear equations provides the coordinates of the point of intersection of the lines represented by the equations.