Problem 79 - Algebra 1: Concepts and Skills

Question

Decide whether the ordered pair is a solution of the system of linear equations. $$ \begin{aligned} &2 x+4 y=2\\\ &-x+5 y=13 \quad(-3,2) \end{aligned} $$

Step-by-Step Solution

Answer

Yes, the ordered pair (-3, 2) is a solution for the given system of linear equations.

Step 1: Substituting the ordered pair into the first equation

Here, \(x = -3\) and \(y = 2\). Substitute these values into the first equation \(2x + 4y = 2\). This gives us \(2 \cdot -3 + 4 \cdot 2 = -6 + 8 = 2\). The left hand side is equivalent to the right hand side, hence, the ordered pair satisfies the first equation.

Step 2: Substituting the ordered pair into the second equation

Using the same values \(x = -3\) and \(y = 2\), substitute in the second equation \(-x + 5y = 13\). This results in \(-(-3) + 5 \cdot 2 = 3 + 10 =13\). The left hand side is equivalent to the right hand side, hence, the ordered pair satisfies the second equation.

Step 3: Conclusions

Since, the ordered pair (-3, 2) satisfies both equations, it is a solution to the system of linear equations given.

Key Concepts

Ordered Pairs in Linear Systems

Understanding how ordered pairs function within linear systems is crucial for grasping the basics of algebra. An ordered pair, typically represented as \( (x, y) \) in a two-dimensional plane, corresponds to the values on the x (horizontal) and y (vertical) axes. In systems of linear equations, these pairs represent potential solutions where the equations in the system intersect.

For example, if we want to determine whether the ordered pair \( (-3, 2) \) is a solution for a given system, we plug these values into the respective equations and check for equality. If the pair satisfies all equations—meaning they produce true statements—it's a solution to the system. Thus, the concept of ordered pairs is an essential part of solving linear systems and analyzing the intersections of lines that represent the equations graphically.

Substitution Method

The substitution method is one of the strategies used to find solutions for systems of linear equations. It involves replacing one variable with an expression derived from another equation to solve for one unknown at a time.

For instance, if you have two equations \( y = 2x \) and \( y = 5 - x \), you can substitute the expression from the first equation into the second, resulting in the equation \( 2x = 5 - x \). This equation can now be solved to find the value of \( x \) and subsequently the value of \( y \). It's a straightforward method that often simplifies the process of finding the intersection of the lines, hence the solutions. In our exercise, substitution isn't necessary as we are verifying whether a given ordered pair is a solution, but in other problems, this technique is key.

Algebraic Solutions

Algebraic solutions involve manipulating equations using algebraic methods to find the values of unknown variables. This can be done using a variety of methods such as substitution, elimination, and graphing. The goal is to either solve for one variable in terms of another and then substitute back to find the definitive solution or to manipulate the equations to eliminate one variable and solve the resulting univariate equation.

In the context of our exercise, algebraic solution refers to the steps taken to substitute the provided ordered pair into the system's equations and verify the veracity of the solution. This algebraic verification is a concrete application of algebra, translating abstract equations into relatable intersections on a graph represented by ordered pairs. Simplifying and solving equations are intrinsic to understanding and solving algebraic problems.