Problem 11 - Algebra 1

Question

Decide whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{ll} 3 x-2 y=11 & (5,2) \\ -x+6 y=7 \end{array} $$

Step-by-Step Solution

Answer

After performing all the steps, it is concluded that (5,2) is not a solution to the system of equations because the right side of both equations is not equal to the left side after the substitution.

Step 1: Substitute in Equation 1

Substitute \(x = 5\) and \(y = 2\) into the first equation \(3x – 2y = 11\). The left-hand side therefore becomes \(3(5) - 2(2) = 11.\)

Step 2: Substitute in Equation 2

Substitute \(x = 5\) and \(y = 2\) into the second equation \(-x + 6y = 7\). The left-hand side therefore becomes \(-5 + 6(2) = 7.\)

Step 3: Check Equality for Both Equations

If after substituting the values, both equations are found true then the ordered pair (5,2) is a solution to the system of equations, otherwise not.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed on a coordinate plane. Each equation in a system typically takes the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) and \(y\) are variables. The key attribute of linear equations is that their graph is always a line.

These equations can represent various relationships in real-world scenarios, such as financial planning or motion along a straight path. In systems of equations, linear equations are usually paired to find a common solution that satisfies all equations involved.

• The number of solutions to a system of linear equations can be zero, one, or infinitely many.
• Graphically, no solution means the lines are parallel; one solution is where they intersect at a single point; infinitely many solutions occur when the lines overlap entirely.

Understanding the nature of linear equations is fundamental when working with systems of equations. It helps students predict whether a solution is possible through methods like substitution or elimination, which we'll explore more in the sections below.

The Concept of Solutions in Equations

Solutions of equations refer to the set of values that satisfy all equations in a given system simultaneously. When we solve a system of linear equations, we're looking for values of the variables that make all the equations true at the same time.

Let's consider the problem at hand; by substituting \(x = 5\) and \(y = 2\) into the equations, we verify if these values satisfy both expressions:

• Substitute into the first equation: \(3x - 2y = 11\). Plugging in the values gives \(3(5) - 2(2) = 15 - 4 = 11\) which is correct.
• Substitute into the second equation: \(-x + 6y = 7\). Plugging in these values results in \(-5 + 6(2) = -5 + 12 = 7\) which checks out as well.

Since both equations are satisfied with \(x = 5\) and \(y = 2\), the ordered pair \((5,2)\) is a solution to this system of linear equations.

The solutions for systems of equations can sometimes be multiple points, or even no points at all, depending on how the equations interact when visualized graphically. This concept is key to understanding whether it is feasible for a given set of ordered pairs to be a solution.

Ordered Pairs in Systems of Equations

An ordered pair, written as \((x, y)\), is a set of values corresponding to the variables in equations of systems. Ordered pairs are essential as they represent potential solutions in the system of equations and help visualize points on a graph.

When checking if an ordered pair is a solution to a system of equations, you plug in the \(x\) and \(y\) components into all equations of the system. If the ordered pair satisfies each equation after substitution, then it is considered a solution.

• Ordered pairs often reflect real-world situations; think coordinates on a map indicating a specific location.
• The solution \((5, 2)\) being correct in our exercise means it lies at the intersection point of the lines represented by the given linear equations.

Understanding ordered pairs in the context of systems of equations makes it easier to identify solutions and interpret them in practical applications. This often involves trial and error, especially in more complex systems, but mastering this core concept is at the foundation of effective equation solving.