"3x + 2y = 7" represents a simple linear equation in two variables, a fundamental concept in algebra with far-reaching applications in various fields. Understanding this equation and its solutions is crucial for grasping more complex mathematical concepts and solving real-world problems involving relationships between two variables. This article will explore this seemingly simple equation in a question-and-answer format, revealing its nuances and applications.
I. What does the equation 3x + 2y = 7 represent?
The equation 3x + 2y = 7 represents a linear relationship between two variables, x and y. It signifies that for any pair of values (x, y) that satisfy this equation, three times the value of x plus two times the value of y will always equal 7. Geometrically, this equation represents a straight line on a Cartesian coordinate plane. Each point on the line represents a solution to the equation.
II. How do we find solutions to the equation 3x + 2y = 7?
There are infinitely many solutions to this equation. We can find some solutions by assigning a value to one variable and solving for the other.
Method 1: Substitution: Let's say we choose x = 1. Substituting this into the equation gives: 3(1) + 2y = 7. Solving for y, we get 2y = 4, therefore y = 2. So (1, 2) is one solution. Similarly, if we let x = 3, we get 3(3) + 2y = 7, which simplifies to 2y = -2, giving y = -1. Thus (3, -1) is another solution.
Method 2: Rearrangement and Table of Values: We can rearrange the equation to solve for one variable in terms of the other. Let's solve for y: 2y = 7 - 3x => y = (7 - 3x)/2. Now we can create a table of values:
III. What does the graph of 3x + 2y = 7 look like?
The graph of 3x + 2y = 7 is a straight line. We can plot the solutions we found earlier – (1, 2), (3, -1), (-1, 5) – on a Cartesian coordinate system. Connecting these points will reveal a straight line. The line represents all possible solutions to the equation. The slope of the line is -3/2 (change in y over change in x), and the y-intercept (the point where the line crosses the y-axis) is 3.5 (found by setting x = 0).
IV. What are the real-world applications of this type of equation?
Linear equations like 3x + 2y = 7 have numerous real-world applications:
Mixture Problems: Imagine mixing two types of coffee beans, one costing $3 per pound (x) and another costing $2 per pound (y) to create a 7-pound blend. The equation 3x + 2y = 7 represents the total cost of the blend.
Resource Allocation: A company might have 7 hours of machine time available (7). Task A takes 3 hours per unit (x) and Task B takes 2 hours per unit (y). The equation helps determine the possible combinations of units for A and B that can be produced within the time constraint.
Budgeting: Suppose you have a budget of $7. Apples cost $3 each (x) and oranges cost $2 each (y). The equation models the different combinations of apples and oranges you can buy.
V. How can we solve this equation with another equation to find a unique solution?
A single linear equation in two variables has infinitely many solutions. To find a unique solution, we need a second linear equation involving the same variables (a system of equations). For example, if we have the system:
3x + 2y = 7
x - y = 2
We can solve this system using substitution or elimination methods to find the unique values of x and y that satisfy both equations simultaneously. In this case, the solution would be x = 2 and y = 0.
Takeaway: The seemingly simple equation 3x + 2y = 7 is a powerful tool for understanding linear relationships and solving real-world problems. Its graphical representation as a straight line and the multiple methods for finding its solutions highlight its versatility and importance in mathematics and beyond.
FAQs:
1. Can this equation be solved graphically without plotting multiple points? Yes, you can find the x-intercept (by setting y = 0) and the y-intercept (by setting x = 0). These two points define the line.
2. What if the equation is more complex, with fractions or decimals? The same principles apply. You can simplify the equation by multiplying through by a common denominator or clearing decimals before solving.
3. What happens if the system of equations has no solution or infinitely many solutions? This occurs when the lines representing the equations are parallel (no solution) or coincident (infinitely many solutions).
4. How can I use software or calculators to solve these equations? Many graphing calculators and software packages (like MATLAB, Python with NumPy/SciPy) can solve linear equations and systems of equations quickly and efficiently.
5. Are there other ways to represent this linear relationship besides an equation? Yes, you can represent it using a table of values, a graph, or even a verbal description of the relationship between x and y.
Note: Conversion is based on the latest values and formulas.
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