Question

In each case verify that the given values satisfy (b) \(x=4, y=3\) satisfy \(x+y=7\) and the given simultaneous equations: (a) \(x=2, y=-2\) satisfy \(7 x+y=12\) and (c) \(x=-3, y=2\) satisfy \(8 x-y=-26\) \(-3 x-y=-4\) and \(9 x+2 y=-23\)

Short answer

a) \(7x + y = 12\) when \(x = 2\) and \(y = -2\) b) \(x + y = 7\) when \(x = 4\) and \(y = 3\) c) Does \(x = -3, y = 2\) satisfy the following linear system: \[ \begin{cases} 8x - y = -26 \\ -3x - y = -4 \\ 9x + 2y = -23 \end{cases} \] Answer: a) Yes, the given values satisfy the equation \(7x + y = 12\). b) Yes, the given values satisfy the equation \(x + y = 7\). c) The given values satisfy the first and third equations, but not the second equation.

Step-by-step solution

Part (a) Verification

Plug the given values of x and y into the equation \(7x + y = 12\): \[7(2) + (-2) = 14 - 2 = 12\] Since both sides are equal, the given values \(x = 2, y = -2\) satisfy the equation \(7x + y = 12\).

Part (b) Verification

Plug the given values of x and y into the equation \(x + y = 7\): \[4 + 3 = 7\] Since both sides are equal, the given values \(x = 4, y = 3\) satisfy the equation \(x + y = 7\).

Part (c) Verification

First, we will plug the given values of x and y into the equation \(8x - y = -26\): \[8(-3) - 2 = -24 - 2 = -26\] Since both sides are equal, the given values satisfy the first equation. Now, plug the given values of x and y into the equation \(-3x - y = -4\): \[-3(-3) - 2 = 9 - 2 = 7\] Since both sides are not equal, the given values \(x = -3, y = 2\) do not satisfy the second equation \(-3x - y = -4\). Lastly, plug the given values of x and y into the equation \(9x + 2y = -23\): \[9(-3) + 2(2) = -27 + 4 = -23\] Since both sides are equal, the given values satisfy the third equation. In summary, the given values \(x = -3, y = 2\) satisfy the first and third equations, but not the second equation.

Key concepts

Mathematical Verification

Mathematical verification is a critical part of solving any mathematical problem. It involves ensuring that any proposed solution actually meets the conditions set by the original equations. In the context of simultaneous equations, verification is done by substituting the solution values back into the original equations. When we look at making sure that \(x=2, y=-2\) satisfy \(7x + y = 12\), we replace \(x\) with 2 and \(y\) with -2. By calculating \(7(2) + (-2)\) and reaching an answer of 12, we verify that this solution is indeed correct since the answer aligns with the right side of the equation. Verification is a powerful tool to confirm the accuracy of the answer and ensure a full understanding of the solution.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. For example, in the given exercise, equations like \(8x - y = -26\) and \(9x + 2y = -23\) are part of a system. The objective is to find a common solution—a set of variable values that makes all equations true simultaneously. There are various methods to solve these systems including graphing, substitution, elimination, and matrix operations.

In real-world terms, such systems can represent multiple relationships or constraints that need to be met at the same time, such as budgeting, mixing ingredients, or planning schedules. The ability to solve these systems is fundamental in many fields including mathematics, physics, engineering, economics, and so on.

Algebraic Solution

An algebraic solution is a way to find the answer to an equation or system of equations using algebraic manipulations. This involves operations like addition, subtraction, multiplication, division, and applying the distributive property. For simultaneous equations, we often manipulate the equations algebraically to isolate one variable and then substitute back into another equation. This process, when applied to find \(x=4, y=3\) as the solutions for \(x + y = 7\), exemplifies how an algebraic solution works—through logical, step-by-step operations that simplify and solve the equations. Teaching the method of algebraic solutions leads students to develop a deeper understanding of the underlying principles of algebra and nurtures their problem-solving skills.

Equations Solving

Equations solving is the act of determining the values of variables that make an equation or a system of equations true. For single equations like \(x + y = 7\), it is a matter of straightforward algebra. However, when dealing with simultaneous equations, such as \(x=2, y=-2\) satisfying both \(7x + y = 12\) and \(x + y = 7\), it becomes more complex as we need a solution that works for all equations at once. There are a few key strategies to solve these: graphing, where you find where two lines intersect; substitution, where you solve one equation for one variable and plug it into the other; and elimination, where you add or subtract equations to eliminate a variable. Each strategy requires a strong grasp of algebraic principles and critical thinking.