Problem 81 - Algebra 1: Concepts and Skills

Question

Decide whether the ordered pair is a solution of the system of linear equations. $$ \begin{aligned} 8 x+4 y &=6 \\ 4 x+y &=3 \end{aligned} \quad(1,-1) $$

Step-by-Step Solution

Answer

The ordered pair (1, -1) is not a solution to the system of linear equations.

Step 1: Substitute (1,-1) into the first equation

Replace x with 1 and y with -1 in the first equation \(8x+4y = 6\). You will then solve the equation to determine if it holds true.

Step 2: Substitute (1,-1) into the second equation

Next, substitute x with 1 and y with -1 in the second equation \(4x+y = 3\). After that, solve the equation to see if it is true.

Step 3: Evaluate if the pair is a solution

Once both equations have been solved with the specific substituted values, determine if the pair satisfies both equations. If true for both, then the pair is indeed a solution for the system of linear equations.

Key Concepts

Understanding Ordered Pairs

In mathematics, an ordered pair is a fundamental concept used to represent the values of two variables together. It is written in the form

• \( (x, y) \)

The position of each element in the pair matters, with the first value representing \( x \) and the second one \( y \).
Think of it as a way to plot points on a graph, where \( x \) is the horizontal position and \( y \) the vertical.
When dealing with systems of linear equations, an ordered pair like \( (1, -1) \) suggests potential values for the variables that could satisfy each equation in the system.
By substituting these values into each equation, we can test if the pair is a true solution.

Exploring the Substitution Method

The substitution method is a hands-on approach used in solving systems of linear equations.
It involves replacing one variable with a numerical value or another expression, which simplifies the equations and helps in finding solutions.
For instance, using our given ordered pair \((1, -1)\), we replace \( x \) and \( y \) in each equation.

• First equation: Substitute 1 for \( x \) and -1 for \( y \) in \( 8x + 4y = 6 \).
• Second equation: Again, substitute 1 for \( x \) and -1 for \( y \) in \( 4x + y = 3 \).

After substitution, solve the equations individually to verify if the expressions equate as expected.
This confirms if the ordered pair is a valid solution for the system.

Verifying Solutions of Equations

Solutions to systems of linear equations are verified by checking if the ordered pair satisfies all given equations simultaneously.
To do this, perform a substitution with each pair element and simplify the resulting equations.

• For example, in the equation \( 8x + 4y = 6 \), substitute \( x = 1 \) and \( y = -1 \), and calculate to see if it equals 6.
• Similarly, substitute in \( 4x + y = 3 \) and check for equality.

If both conditions hold true, the pair is indeed a solution.
In our exercise, the pair \( (1, -1) \) must equate both expressions for the system to be true.
If any equation fails, the pair isn't a solution, making this verification step crucial.