Problem 83
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
Verified Answer
The point \((4,-1)\) does satisfy the equation \(x + 2y - 2 = 0\), but it does not satisfy the equation \(x - 2y - 6 = 0\).
1Step 1: Substitute into the first equation
When \(x = 4\) and \(y = -1\) are substituted into \(x + 2y - 2\), we get \(4 + 2(-1) - 2\). Next, simplify that expression.
2Step 2: Simplify the first equation
After substituting and simplifying, we get \(4 - 2 - 2 = 0\). Hence, the equation \(x + 2y - 2 = 0\) is satisfied for the point \((4,-1)\).
3Step 3: Substitute into the second equation
When \(x = 4\) and \(y = -1\) are substituted into \(x - 2y - 6\), we get \(4 - 2(-1) - 6\). Next, simplify that expression.
4Step 4: Simplify the second equation
After substituting and simplifying, we get \(4 + 2 - 6 = 0\). Hence, the equation \(x - 2y - 6 = 0\) is not satisfied for the point \((4,-1)\), because the result is \(0\), not \(6\).
Key Concepts
Coordinate PairsSubstitution MethodEquation SimplificationSystems of Equations
Coordinate Pairs
One of the fundamental elements of algebra and geometry is the concept of coordinate pairs. They are used to represent positions on a grid, typically drawn as an 'x' and 'y' axis to form a coordinate plane. Each pair consists of two numbers, which are referred to as coordinates, where the first number corresponds to the position along the x-axis (horizontal) and the second to the y-axis (vertical).
For instance, the pair \( (4,-1) \) indicates a point that is located 4 units to the right of the origin along the x-axis, and 1 unit down from the origin along the y-axis. Testing whether a coordinate pair satisfies an equation is an essential skill in solving and understanding systems of equations.
For instance, the pair \( (4,-1) \) indicates a point that is located 4 units to the right of the origin along the x-axis, and 1 unit down from the origin along the y-axis. Testing whether a coordinate pair satisfies an equation is an essential skill in solving and understanding systems of equations.
Substitution Method
When solving systems of equations, the substitution method is a vital tool. It involves replacing one variable with an equivalent expression from another equation. This method simplifies the system to a single variable, making the equations easier to solve.
To apply the substitution method, you must first solve one of the equations for one variable in terms of the others. Then you substitute this expression into the other equation, thus reducing the number of variables and enabling you to find the solution more straightforwardly.
To apply the substitution method, you must first solve one of the equations for one variable in terms of the others. Then you substitute this expression into the other equation, thus reducing the number of variables and enabling you to find the solution more straightforwardly.
Equation Simplification
Simplification of an equation is a crucial step in solving mathematical problems. It involves reducing the complexity of an equation to more easily discernible terms. The process typically includes combining like terms, which are terms that have the same variable raised to the same power, and performing basic arithmetic operations such as addition, subtraction, multiplication, and division.
Take the expression \(4 + 2(-1) - 2\), for example. Simplifying starts with handling the multiplication, giving us \(4 - 2 - 2\), and further adding and subtracting to break it down to \(0\). Simplification helps in cleanly presenting whether a coordinate pair is a solution to an equation.
Take the expression \(4 + 2(-1) - 2\), for example. Simplifying starts with handling the multiplication, giving us \(4 - 2 - 2\), and further adding and subtracting to break it down to \(0\). Simplification helps in cleanly presenting whether a coordinate pair is a solution to an equation.
Systems of Equations
Systems of equations consist of two or more equations that share two or more variables. The solutions to these systems are the values of the variables that satisfy all included equations simultaneously. These systems can be solved by various methods, such as graphing, substitution, or elimination. Each method has its advantages and appropriate contexts.
In the context of the exercise provided, by substituting the coordinate pair \( (4,-1) \) into both equations and simplifying, we can ascertain whether the given point is a solution to each equation in the system. If the point satisfies all equations in the system, it can be considered a solution to the system as a whole.
In the context of the exercise provided, by substituting the coordinate pair \( (4,-1) \) into both equations and simplifying, we can ascertain whether the given point is a solution to each equation in the system. If the point satisfies all equations in the system, it can be considered a solution to the system as a whole.
Other exercises in this chapter
Problem 82
a. Graph \(y-e^{x}\) and \(y-1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y-e^{x}\) and \(y-1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same
View solution Problem 83
Evaluate or simplify each expression without using a calculator. $$ \log 10^{7} $$
View solution Problem 83
In Exercises \(83-88,\) let \(\log _{b} 2=A\) and \(\log _{b} 3=\) C. Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \frac{3}{2} $$
View solution Problem 83
Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions.
View solution