Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Determine whether each point is a solution of the equation. Equation Points \(y=2 x^{2}-7 x+3\) (a) \((1,-1)\) (b) \((3,0)\)

Step-by-Step Solution

Answer

Point (1, -1) is not a solution and point (3, 0) is a solution to the given equation.

Step 1: Determine if (1, -1) is a solution

Substitute \(x = 1\) and \(y = -1\) into the equation: \(-1 = 2*(1)^2 -7*1 + 3\). Simplify to find if the equality holds: \(-1 = 2 - 7 + 3 = -2\). The equality does not hold, so (1, -1) is not a solution.

Step 2: Determine if (3, 0) is a solution

Substitute \(x = 3\) and \(y = 0\) into the equation: \(0 = 2*(3)^2 -7*3 + 3\). Simplify to find if the equality holds: \(0 = 2*9 - 21 + 3 = 18 - 21 +3 = 0\). The equality holds, so (3, 0) is a solution.

Key Concepts

Algebraic Solutions

Algebraic solutions involve solving equations through algebraic manipulations, using mathematical rules and properties. When dealing with quadratic equations like \(y = 2x^2 - 7x + 3\), the goal is to determine if a given point lies on the parabola represented by the equation.

To check if a point is a solution, substitute the coordinates of the point \((x, y)\) into the equation. Check if both sides of the equation are equal after substitution.

• If they are equal, the point is said to be a solution or satisfies the equation.
• If not, the point does not lie on the curve defined by the equation and is not a solution.

This process requires careful simplification to ensure accuracy.

Coordinate Points

Coordinate points are simply a pair of numbers that define a specific location on a graph, typically written as \((x, y)\). In the context of identifying points that satisfy a quadratic equation, these points are solution candidates that we test by substituting into the equation.

Initially, consider each coordinate point one by one:

• Take the \(x\)-value from the point and substitute it into the equation. This allows us to calculate the value on the right-hand side of the equation.
• Compare this value to the \(y\)-value given in the point.

This comparison helps us determine if the point lies on the curve of the equation. The factors of being right on the curve make a point a valid solution.

Substitution Method

The substitution method is a straightforward technique used to determine if a given point solves a given equation. When we use the substitution method in the context of quadratic equations, we replace the variable \(x\) with the \(x\)-coordinate from the point and \(y\) with the \(y\)-coordinate.

For example, if you are evaluating the point \((1, -1)\) for the equation \(y = 2x^2 - 7x + 3\), execute the following steps:

• Replace \(x\) with 1 and \(y\) with -1.
• Calculate: \(-1 = 2(1)^2 - 7(1) + 3\). Simplify to see if both sides match.

This method essentially cross-checks if both sides of the equation hold true based on the given point.

If they equate perfectly, then the substitution shows that the point is on the graph of the equation. If there is a discrepancy, the point does not match the equation's graph.