Write A System Of Equations With The Solution 4

How to Write a System of Equations with a Specific Solution: A Complete Guide

Understanding how to construct a system of equations that yields a predetermined solution is a fundamental skill in algebra that bridges abstract concepts with practical problem-solving. At its core, this task involves reverse-engineering mathematical statements. When we say we want a system with the solution (4, 2), we are specifying that the ordered pair where x = 4 and y = 2 must satisfy every equation in the system simultaneously. This means that when you substitute 4 for x and 2 for y into each equation, the resulting mathematical statement must be true (e.g., 8 = 8, 10 = 10). The power of this exercise lies in its demonstration of the solution concept—a solution is not just a number pulled from thin air, but a point of perfect harmony where multiple conditions are met at once. This guide will move you from the basic "how" to the deeper "why," equipping you with the strategies to create infinite valid systems for any given point.

Detailed Explanation: Deconstructing the Task

A system of equations is simply a set of two or more equations that share the same variables. The solution to the system is the specific value(s) for those variables that make all equations true at the same time. Graphically, for two variables in two dimensions (x and y), each linear equation represents a line. The solution is the precise coordinate point where these lines intersect. Therefore, our goal is to design equations whose lines (or curves) all pass through the chosen point, (4, 2).

The process is elegantly simple in principle: start with your target solution, (x₁, y₁), and build equations that are satisfied by plugging in x = x₁ and y = y₁. The creative freedom comes from how you construct those equations. You can create equations of any degree (linear, quadratic), with any coefficients, as long as the point lies on the curve. This isn't about finding a single correct answer; it's about understanding the infinite family of correct answers and the principles that unite them.

Step-by-Step Breakdown: Building Your System

Let’s systematically create a system with the solution (4, 2). We will create two equations for a standard two-variable system.

Step 1: Anchor with the Solution. First, internalize that our anchor point is x = 4, y = 2. Every equation we write must work when these values are substituted.

Step 2: Create the First Equation. The simplest approach is to start with a linear equation (y = mx + b). We need to choose a slope (m) and a y-intercept (b) so the line passes through (4, 2).

• Choose any slope. Let’s pick m = 3.
• Use the point-slope form: y - y₁ = m(x - x₁).
• Plug in: y - 2 = 3(x - 4).
• Simplify to slope-intercept form: y - 2 = 3x - 12 → y = 3x - 10.
• Verification: Substitute x=4: y = 3(4) - 10 = 12 - 10 = 2. ✓ Correct.

Step 3: Create the Second Equation (With Variety). To show the breadth of possibilities, let’s make the second equation non-linear (a parabola). A simple quadratic form is y = ax² + bx + c.

• We need a, b, and c such that when x=4, y=2.
• Choose a = 1 (for simplicity).
• Choose b = 0 (another choice to reduce variables).
• Now solve for c: 2 = (1)(4)² + (0)(4) + c → 2 = 16 + c → c = -14.
• Our second equation is: y = x² - 14.
• Verification: x=4: y = 16 - 14 = 2. ✓ Correct.

Step 4: Assemble and Verify the System. Our final system is:

1. y = 3x - 10
2. *y = x² - 14