Which Ordered Pair Is a Solution of the System x + 4y = 19, y = -2x – 4?
In the field of mathematics, one encounters various systems of equations that require solving for unknown variables. One such system is given by the equations x + 4y = 19 and y = -2x – 4. In this article, we will explore how to find the ordered pair that satisfies this system and delve into the intricacies of solving such equations. Additionally, a FAQs section will address common queries related to this topic.
Solving the System of Equations:
To determine the ordered pair that satisfies the given system of equations, we need to substitute the value of y from the second equation into the first equation. Let’s start by substituting y = -2x – 4 into x + 4y = 19:
x + 4(-2x – 4) = 19
x – 8x – 16 = 19
-7x – 16 = 19
-7x = 35
x = -5
Now that we have found the value of x, we can substitute it back into one of the original equations to find the corresponding value of y. Let’s substitute x = -5 into y = -2x – 4:
y = -2(-5) – 4
y = 10 – 4
y = 6
Therefore, the ordered pair (-5, 6) is a solution to the given system of equations.
Q1. Can I solve a system of equations using a graph?
A1. Yes, graphing is another method to solve systems of equations. By plotting the equations on a coordinate plane, the point of intersection represents the solution. However, this method might not be as accurate or efficient as algebraic methods.
Q2. Are there other methods to solve systems of equations?
A2. Yes, apart from substitution, another popular method is called elimination. In elimination, you eliminate one variable by adding or subtracting the equations so that one variable cancels out, allowing you to solve for the other variable.
Q3. What if the system has no solution?
A3. If the system of equations has no solution, it means that the lines representing the equations are parallel and will never intersect. In this case, the system is said to be inconsistent.
Q4. What if the system has infinitely many solutions?
A4. If the system of equations has infinitely many solutions, it means that the lines representing the equations are coincident, and every point on one line is also on the other line. In this case, the system is said to be dependent.
Q5. Can I solve a system of equations with more than two variables?
A5. Yes, the methods used to solve systems of equations with two variables can also be applied to systems with more than two variables. However, the calculations become more complex as the number of variables increases.
Solving systems of equations is a fundamental concept in mathematics, allowing us to find the values of unknown variables. In the case of the system x + 4y = 19 and y = -2x – 4, we found that the ordered pair (-5, 6) satisfies both equations. By using substitution, we were able to determine the values of x and y that make the system true. Remember that there are alternative methods, such as graphing and elimination, to solve systems of equations.