Problem 45
Question
\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{array}{l}{0.4 x+1.2 y=14} \\ {12 x-5 y=10}\end{array}\right. $$
Step-by-Step Solution
Verified Answer
The solution to the system is \((5, 10)\).
1Step 1: Eliminate Decimals
To eliminate the decimal in the first equation, let's multiply the entire equation by 10, resulting in the equation: \( 4x + 12y = 140 \).
2Step 2: Simplify the System of Equations
The system now consists of the following equations: \( 4x + 12y = 140 \) and \( 12x - 5y = 10 \).
3Step 3: Use the Elimination Method
We will multiply the first equation by 3 to align it for elimination: \( 12x + 36y = 420 \). Now, subtract the second equation \( 12x - 5y = 10 \) from the modified first equation to eliminate \( x \).
4Step 4: Eliminate and Solve for y
Subtracting the equations gives \( 0 + 41y = 410 \), which simplifies to \( y = 10 \).
5Step 5: Substitute Back to Find x
Substitute \( y = 10 \) back into the first simplified equation: \( 4x + 12(10) = 140 \), which simplifies to \( 4x = 20 \). Solve for \( x \) by dividing both sides by 4, resulting in \( x = 5 \).
6Step 6: Write the Solution as an Ordered Pair
The solution to the system of equations is \((x, y) = (5, 10)\). This is the point where the lines represented by the equations intersect.
Key Concepts
Elimination MethodOrdered PairsSolving Linear Equations
Elimination Method
The elimination method is a strategic way of solving systems of linear equations. It involves adding or subtracting equations in order to eliminate one of the variables. This makes it easier to solve for the remaining variable. Here’s how it’s done:
In our example, we used the elimination method by aligning equations so that the coefficients of one of the variables matched. By manipulating the equations in this way, we were able to elegantly eliminate the variable and solve for the other.
- First, decide which variable you want to eliminate. This usually depends on the coefficients of the variables.
- Multiply one or both of the equations by a necessary factor so that the coefficients of the chosen variable are equal or opposites.
- Add or subtract the equations to eliminate the chosen variable.
- Solve the resulting equation for the remaining variable.
- Substitute the value of the found variable back into one of the original equations to solve for the other variable.
In our example, we used the elimination method by aligning equations so that the coefficients of one of the variables matched. By manipulating the equations in this way, we were able to elegantly eliminate the variable and solve for the other.
Ordered Pairs
In mathematics, an ordered pair is a set of numbers used to locate a point in the coordinate plane. It consists of two elements: a first and a second coordinate, usually represented as \(x, y\). This notation signifies a specific position in a two-dimensional space.
Ordered pairs are especially prominent when dealing with solutions to systems of equations. In the context of solving systems of equations, finding an ordered pair means identifying the point where two lines intersect, representing the solution for both equations.
For example, the ordered pair \(5, 10\) we found indicates that \(x = 5\) and \(y = 10\) satisfy both equations. This means that if you were to plot the two lines represented by our equations on a graph, they would intersect at the point \(5, 10\).
Ordered pairs are especially prominent when dealing with solutions to systems of equations. In the context of solving systems of equations, finding an ordered pair means identifying the point where two lines intersect, representing the solution for both equations.
For example, the ordered pair \(5, 10\) we found indicates that \(x = 5\) and \(y = 10\) satisfy both equations. This means that if you were to plot the two lines represented by our equations on a graph, they would intersect at the point \(5, 10\).
Solving Linear Equations
Solving linear equations is a fundamental skill in algebra where you find the value of variables that satisfy the equation. These equations represent straight lines and typically involve constants and variables.
Key steps to solve a linear equation include:
In systems of equations, each linear equation provides a constraint involving the same variables. Solving involves finding values that satisfy all constraints simultaneously, leading to solutions like an intersecting point on a graph, as seen in this example. By arranging the system into simpler equations, we successfully determined \(x = 5\) and \(y = 10\).
Key steps to solve a linear equation include:
- Isolate the variable on one side of the equation. This often involves basic operations such as addition, subtraction, multiplication, or division.
- Simplify both sides of the equation as much as possible.
- Perform the same operation on both sides to maintain the balance of the equation, eventually solving for the variable.
In systems of equations, each linear equation provides a constraint involving the same variables. Solving involves finding values that satisfy all constraints simultaneously, leading to solutions like an intersecting point on a graph, as seen in this example. By arranging the system into simpler equations, we successfully determined \(x = 5\) and \(y = 10\).
Other exercises in this chapter
Problem 45
Find the inverse of the matrix. For what value(s) of x, if any, does the matrix have no inverse? $$ \left[\begin{array}{rrr}{1} & {e^{x}} & {0} \\ {e^{x}} & {-e
View solution Problem 45
Flight of a Rocket A hill is inclined so that its "slope" is \(\frac{1}{2},\) as shown in the figure. We introduce a coordinate system with the origin at the ba
View solution Problem 45
Determine \(A\) and \(B\) in terms of \(a\) and \(b\) $$ \frac{a x+b}{x^{2}-1}=\frac{A}{x-1}+\frac{B}{x+1} $$
View solution Problem 46
Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} 2 x-5 y =4 \\ x+y-z =8 \\ 3 x +5 z=0 \end{aligned}\right. $$
View solution