Problem 83 - College Algebra Essentials

Question

Exercises \(83-85\) will help you prepare for the material covered in the first section of the next chapter. a. Does \((4,-1)\) satisfy \(x+2 y=2 ?\) b. Does \((4,-1)\) satisfy \(x-2 y=6 ?\)

Step-by-Step Solution

Answer

Yes, the point \((4,-1)\) satisfies both the equations \(x+2 y=2\) and \(x-2 y=6\).

Step 1: Substitute into the first equation

Substitute the pair \((4,-1)\) into the first equation \(x+2 y=2\). So, it becomes \(4 + 2*(-1) = 2\)

Step 2: Simplify the left side

Simplify the equation to find the left-hand side value. \(4 + 2*(-1)\) simplifies to \(4 - 2 = 2\). Both sides match, therefore the point \((4,-1)\) satisfy the equation \(x+2 y=2\).

Step 3: Substitute into the second equation

Now substitute the pair \((4,-1)\) into the second equation \(x-2 y=6\). So, it becomes \(4 - 2*(-1) = 6 \)

Step 4: Simplify the left side

Simplify the equation to find the left-hand side value. \(4 - 2*(-1)\) simplifies to \(4 + 2 = 6\). Both sides match, therefore the point \((4,-1)\) satisfy the equation \(x-2 y=6\).

Key Concepts

Substitution Method

The substitution method is a handy tool for solving systems of equations, especially linear ones. In this approach, you substitute the values you need to check directly into the equation. This method is all about replacing the variables with specific values and checking if the equation holds, or in other words, if both sides are equal.
For example, when you have the equation \(x + 2y = 2\), you can check if a specific point, say \((4,-1)\), satisfies it by substituting \(x = 4\) and \(y = -1\) into the equation.

• Start by replacing \(x\) with \(4\) and \(y\) with \(-1\)
• The equation will then look like this: \(4 + 2(-1) = 2\)

This method essentially tests if the substitution results in a true statement, thus confirming whether the point is a solution to the equation.
Overall, the substitution method is a straightforward technique that is very useful when dealing with simple linear equations or when verifying specific solutions.

Linear Equations

Linear equations are equations of the first degree, meaning they have no exponents higher than one. These equations can represent straight lines when graphed. The general form of a linear equation in two variables is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Linear equations have some key characteristics:

• They graph to straight lines.
• They have no products or powers of variables.
• Each term is either a constant or a product of a constant with a single variable.

In the original exercise, the equations \(x + 2y = 2\) and \(x - 2y = 6\) are both linear equations. Each of them can be solved using methods like graphing, substitution, or elimination for a pair of values \((x, y)\).
Linear equations are incredibly important because they model many real-world situations. This makes them one of the foundational concepts in algebra and mathematics in general.

Satisfying an Equation

A solution to an equation is said to "satisfy" the equation. This means that when the solution is substituted into the equation, both sides of the equation are equal.
To determine if a point satisfies a given equation, you follow these steps:

• Substitute the point's coordinates into the equation.
• Simplify the expression and check if both sides of the equation remain equal.

This is exactly what was done in the original step-by-step solution you saw earlier. When you plugged \((4, -1)\) into the two equations in the exercise, we found that both sides of each equation matched perfectly after simplification. Hence, \((4, -1)\) satisfies both equations.
This process of checking if a point satisfies an equation is crucial in verifying solutions of systems of equations or for identifying certain graph characteristics. It essentially confirms the validity of a solution.