Problem 81 - Algebra 1

Question

Decide whether the ordered pair is a solution of the system. \begin{aligned} &8 a+4 b=6\\\ &4 a+b=3\left(\frac{3}{4}, 0\right) \end{aligned}

Step-by-Step Solution

Answer

Yes, the ordered pair \((\frac{3}{4}, 0)\) is a solution of the system.

Step 1: Substitute Ordered Pair into Equations

The ordered pair \((\frac{3}{4}, 0)\) represents the variable values: \(a=\frac{3}{4}\) and \(b=0\). Substituting these values into both equations of the system will give us: \n For equation 1: \(8a + 4b = 8*(\frac{3}{4}) + 4*0 = 6\). After calculation, we get \(\frac{24}{4}+0 = 6\), which means \(6=6\). Thus, this equation is true and satisfied. \n For equation 2: \(4a + b = 4*(\frac{3}{4}) + 0 =3\). We get after calculation \(\frac{12}{4}+0 = 3\), which also means \(3=3\). Hence, this equation is also true and satisfied.

Step 2: Check if both Equations are Satisfied

Both equations of the system are satisfied when substituting the given ordered pair \((\frac{3}{4}, 0)\). Hence, it confirms that the ordered pair \((\frac{3}{4}, 0)\) is indeed a solution to the system.

Key Concepts

Understanding Ordered Pairs

An ordered pair, typically written in the form \( (x, y) \), is a fundamental concept in the algebraic representation of solutions to systems of linear equations. It consists of two elements, where the first element represents the value of one variable, typically \( x \), and the second element represents the value of the other variable, often \( y \). The order of these numbers is crucial, as the first one corresponds to the horizontal axis (abscissa) and the second to the vertical axis (ordinate) on a coordinate plane.

When analyzing whether an ordered pair is a solution to a system of equations, like the pair \(\left(\frac{3}{4}, 0\right)\) in our exercise, we substitute the values into each equation subtly and meticulously. If, after the substitution, both equations produce true statements, then the ordered pair satisfies the system, and we declare it as a valid solution. This method of verification maintains alignment with our educational ethos of clarity and simplicity.

The Substitution Method Explained

The substitution method is a strategic technique used to find the solution to a system of linear equations. This method involves replacing a variable with its equivalent expression from one equation into the other equation(s). The goal is to isolate one variable and solve the equation for its value.

To embody this process, we look at the values given by the ordered pair and substitute them directly into the equations. Using our exercise, \( a=\frac{3}{4} \) and \( b=0 \) were substituted into the equations. The equations are simplified to show that each side equates, confirming that the system holds true for these values. This pedagogic approach unfolds the algebraic mystery step-by-step and makes the abstract concrete.

Algebraic Solution Deep Dive

Achieving an 'algebraic solution' means solving a system of equations to find the exact numeric values that satisfy all equations simultaneously. This analytical approach is culminated when each step from substitution to simplification reinforces the exactitude of algebra. It involves manipulating and rearranging terms, applying arithmetic operations, and employing critical thinking to join the dots in a meaningful trajectory.

In the scope of our exercise, the algebraic solution is evidenced by the equivalence of both sides of the equations after replacing the variables \( a \) and \( b \) with the respective values from the ordered pair \(\left(\frac{3}{4}, 0\right)\). The equations, when simplified correctly, yielded truthful equalities \(6=6\) and \(3=3\), upholding the algebraic integrity of the solution found. This linear clarity sheds light on the path to computational proficiency.