Solving The System Of Equations 4x = Y And 2x² - Y = 0

In the realm of mathematics, solving a system of equations is a fundamental skill. It involves finding the values of the variables that satisfy all the equations in the system simultaneously. This article delves into the process of solving a specific system of equations, providing a step-by-step approach to arrive at the correct solution set. We will explore the given equations, employ substitution techniques, and verify our solutions to ensure accuracy. Understanding these methods is crucial for various mathematical applications, from algebraic manipulations to graphical interpretations.

Keywords: system of equations, solution set, substitution, algebraic manipulation, quadratic equations, intersection points, graphs of equations, simultaneous equations, variable elimination, solution verification

Understanding the Equations

Before we jump into solving, let's carefully examine the given system of equations:

1. 4x = y
2. 2x² - y = 0

The first equation, 4x = y, is a linear equation. It represents a straight line when graphed on a coordinate plane. The equation tells us that the y-coordinate of any point on this line is four times its x-coordinate. Linear equations are characterized by their constant rate of change, which is represented by the slope.

The second equation, 2x² - y = 0, is a quadratic equation. When rearranged as y = 2x², it represents a parabola opening upwards. Quadratic equations are characterized by their squared term, which results in a curved shape when graphed. The solutions to a quadratic equation can be found by factoring, completing the square, or using the quadratic formula.

To solve this system, we are looking for the points where these two graphs intersect. These intersection points represent the (x, y) pairs that satisfy both equations simultaneously. There are several methods to find these points, and we will utilize the substitution method in this article.

Keywords: linear equation, quadratic equation, parabola, straight line, slope, intersection points, graphing, coordinate plane, substitution method, algebraic methods

The Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with a single variable, which can then be solved using standard algebraic techniques.

In our case, the first equation, 4x = y, is already solved for y. This makes it a perfect candidate for substitution. We can directly substitute the expression 4x for y in the second equation:

2x² - y = 0 becomes 2x² - (4x) = 0

Now we have a quadratic equation in terms of x only: 2x² - 4x = 0

This equation can be solved by factoring. We can factor out a 2x from both terms:

2x(x - 2) = 0

This equation is satisfied if either 2x = 0 or x - 2 = 0.

Solving 2x = 0 gives us x = 0.

Solving x - 2 = 0 gives us x = 2.

So, we have found two possible values for x: x = 0 and x = 2. To find the corresponding y values, we substitute these x values back into either of the original equations. The first equation, 4x = y, is simpler to use for this purpose.

Keywords: substitution method, variable elimination, factoring, quadratic equation, solving for x, solving for y, algebraic techniques, simplification, zero product property, linear equation substitution

Finding the Corresponding y Values

Now that we have the x values, we can find the corresponding y values by substituting them back into the equation 4x = y.

• When x = 0: y = 4 * 0 = 0 So, one solution is the point (0, 0).
• When x = 2: y = 4 * 2 = 8 So, another solution is the point (2, 8).

Therefore, we have found two solution pairs: (0, 0) and (2, 8). These points represent the intersection points of the line 4x = y and the parabola y = 2x².

Keywords: back-substitution, y-coordinate calculation, solution pairs, intersection points, coordinate geometry, linear equation, quadratic equation, verification, simultaneous solutions

Verifying the Solutions

It's always a good practice to verify our solutions by plugging them back into both original equations to ensure they hold true.

• For the solution (0, 0):
  • Equation 1: 4x = y becomes 4 * 0 = 0, which is true.
  • Equation 2: 2x² - y = 0 becomes 2 * 0² - 0 = 0, which is also true.
• For the solution (2, 8):
  • Equation 1: 4x = y becomes 4 * 2 = 8, which is true.
  • Equation 2: 2x² - y = 0 becomes 2 * 2² - 8 = 2 * 4 - 8 = 8 - 8 = 0, which is also true.

Since both solutions satisfy both equations, we can confidently say that they are the correct solutions to the system.

Keywords: solution verification, substitution, equation validation, accuracy check, solution confirmation, algebraic verification, mathematical rigor, problem-solving strategy

The Solution Set

Having verified our solutions, we can now express the solution set. The solution set is the set of all ordered pairs (x, y) that satisfy both equations in the system. In this case, we found two such pairs: (0, 0) and (2, 8).

Therefore, the solution set is {(0, 0), (2, 8)}.

This corresponds to option B in the original question.

Keywords: solution set, ordered pairs, set notation, final answer, problem resolution, mathematical solution, system of equations solution

Conclusion

In this article, we successfully solved the system of equations 4x = y and 2x² - y = 0 using the substitution method. We found the solution set to be {(0, 0), (2, 8)}. This process involved understanding the types of equations, applying algebraic techniques, and verifying our solutions. The ability to solve systems of equations is a crucial skill in mathematics, with applications in various fields such as physics, engineering, and economics. By mastering these techniques, you can confidently tackle more complex mathematical problems and gain a deeper understanding of the relationships between variables.

Keywords: problem-solving, mathematical techniques, algebraic skills, applications of mathematics, system of equations, solution set, substitution method, conclusion, summary, learning outcomes