Question

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} 3 x+2 y &=2 \\ x-7 y &=-30 \\ \end{array}\right. \\ x=-2, y =4 ;(-2,4) \end{array} $$

Short answer

The values \( x = -2 \) and \( y = 4 \) satisfy both equations.

Step-by-step solution

Substitute Values into the First Equation

Substitute \( x = -2 \) and \( y = 4 \) into the first equation: \( 3x + 2y = 2 \). This becomes: \( 3(-2) + 2(4) = 2 \).

Simplify the First Equation

Perform the multiplication and addition: \( 3(-2) + 2(4) = -6 + 8 = 2 \). Since both sides of the equation are equal, the values satisfy the first equation.

Substitute Values into the Second Equation

Substitute \( x = -2 \) and \( y = 4 \) into the second equation: \( x - 7y = -30 \). This becomes: \( -2 - 7(4) = -30 \).

Simplify the Second Equation

Perform the multiplication and addition: \( -2 - 7(4) = -2 - 28 = -30 \). Since both sides of the equation are equal, the values satisfy the second equation.

Conclude the Verification

Since the values \( x = -2 \) and \( y = 4 \) satisfy both equations, \( (-2, 4) \) is a solution to the system of equations.

Key concepts

substitution method

The substitution method is a popular technique for solving systems of equations. It involves replacing one variable with an expression derived from one of the equations. This helps in simplifying the problem and makes it easier to solve. For example, if we have two equations:
1. \(3x + 2y = 2\)
2. \(x - 7y = -30\)
We can solve one of these equations for one variable. Let's solve the second equation for \(x\):
\( x = 7y - 30\)
Next, we substitute this value of \(x\) into the first equation and solve for \(y\). Once \(y\) is determined, we substitute it back to find the value of \(x\). This step-by-step approach helps ensure accuracy and clarity.

linear equations

Linear equations are equations of the first degree, which means that the variables are raised to a power of one. The general form of a linear equation in two variables is \(Ax + By = C\), where A, B, and C are constants. In the given exercise, the equations \(3x + 2y = 2\) and \(x - 7y = -30\) are linear equations. Linear equations can be graphed as straight lines in a coordinate plane.
Each solution of a linear equation corresponds to a point on its line. For systems of linear equations, the solution is the point where the lines intersect. If they intersect at exactly one point, there is one unique solution.

solving systems of equations

Solving systems of equations involves finding values of variables that satisfy all given equations simultaneously. There are multiple methods to solve these systems, including:

• Substitution Method
• Elimination Method
• Graphical Method

In this exercise, the substitution method is used. The main idea is to solve one equation for one variable, then substitute that into another equation. Another common method is the elimination method, which involves adding or subtracting equations to eliminate one of the variables. Each method has its advantages depending on the specific problem.

verification of solutions

Verification of solutions is a crucial step to ensure that the values obtained for the variables truly solve the original system of equations. After finding potential solutions, substitute these values back into the original equations to check if they satisfy both equations. In the given exercise, after substituting \(x = -2\) and \(y = 4\) into both equations:

• First Equation: \(3(-2) + 2(4) = 2\). This simplifies to \(-6 + 8 = 2\), which is correct.
• Second Equation: \(-2 - 7(4) = -30\). This simplifies to \(-2 - 28 = -30\), which is correct.

Since both equations are satisfied, \((-2, 4)\) is indeed the solution to the system. Verification provides confidence that the solution is accurate and correct.