Problem 41
Question
Exercises \(41-43\) will help you prepare for the material covered in the first section of the next chapter. Solve the system: $$ \left\\{\begin{aligned} x+y+2 z &=19 \\ y+2 z &=13 \\ z &=5 \end{aligned}\right. $$
Step-by-Step Solution
Verified Answer
The solution to the system of equations is \(x = 6\), \(y = 3\), and \(z = 5\).
1Step 1 - Identify and start with the simplest equation
The simplest equation to start with is the third equation, \(z = 5\). This equation has only one unknown, z. The value of z is given right away without any need for further solving.
2Step 2 - Substitute z into the second equation to solve for y
The value of z obtained from step 1 can be substituted into the second equation to find the value for y. The second equation is \(y + 2z = 13\). Hence, \(y + 2*5 = 13 => y + 10 = 13\), so \(y = 13 - 10 = 3\). Hence, the value of y is 3.
3Step 3 - Substitute the values of y and z into the first equation to solve for x
Now that we have the values for y and z, these can be substituted into the first equation to find the value of x. The first equation is \(x + y + 2z = 19\). Hence, substituting y and z, we get \(x + 3 + 2*5 = 19 => x + 3 + 10 = 19 => x = 19 - 13 = 6\). So, the value of x is 6.
Key Concepts
Substitution MethodLinear EquationsAlgebraic Solutions
Substitution Method
The substitution method is a powerful algebraic technique used for solving systems of linear equations. It involves replacing one variable with its equivalent based on another equation. The beauty of this method is that it allows the isolation of one variable at a time, reducing the complexity of the problem.
For instance, given a system of equations, we start by identifying the simplest equation, usually one with a single variable. In the case of the problem provided, the third equation, \( z = 5 \), is our starting point. We then use this known value of \( z \) to find \( y \) in the second equation, and subsequently, \( x \) in the first equation. It's a sequential process that involves the following steps:
For instance, given a system of equations, we start by identifying the simplest equation, usually one with a single variable. In the case of the problem provided, the third equation, \( z = 5 \), is our starting point. We then use this known value of \( z \) to find \( y \) in the second equation, and subsequently, \( x \) in the first equation. It's a sequential process that involves the following steps:
- Starting from the simplest equation to find the value of one variable.
- Substituting that value into another equation to determine the second variable.
- Repeating this substitution as necessary until all variables are found.
Linear Equations
Linear equations form the foundation of algebra and appear everywhere in mathematics. They're equations of the first degree, meaning the highest power of the variable is one. Linear equations can have one, two or more variables and are graphically represented as straight lines on a coordinate plane.
The general form of a linear equation with two variables, \( x \) and \( y \), is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants and \( a \) and \( b \) are not both zero. In the context of systems, we have multiple linear equations set equal to different constants but using the same variables. Solving a system of linear equations, such as the one provided:\[\left\{\begin{aligned}x+y+2z &=19 \y+2z &=13 \z &=5 \end{aligned}\right.\]involves finding the values of \( x \), \( y \), and \( z \) that make all equations true simultaneously.
The general form of a linear equation with two variables, \( x \) and \( y \), is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants and \( a \) and \( b \) are not both zero. In the context of systems, we have multiple linear equations set equal to different constants but using the same variables. Solving a system of linear equations, such as the one provided:\[\left\{\begin{aligned}x+y+2z &=19 \y+2z &=13 \z &=5 \end{aligned}\right.\]involves finding the values of \( x \), \( y \), and \( z \) that make all equations true simultaneously.
Algebraic Solutions
Algebraic solutions are the answers obtained by solving equations using algebraic methods—like the substitution method mentioned earlier, the addition (or elimination) method, and by graphical methods as well. These solutions may consist of numbers, expressions, or a set of numbers in the case of systems.
In the example given, the algebraic solution to the system of equations is the set of values for \( x \), \( y \), and \( z \) that satisfy all the equations at the same time. This solution is crucial because it represents the point where all equations in the system intersect, symbolizing the common solution.
The process of finding an algebraic solution often requires a combination of arithmetic operations and logical thinking. It is notably satisfying when we arrive at the concrete values of \( x = 6 \), \( y = 3 \), and \( z = 5 \) that solve the system, providing a clear set of results to a potentially complex problem.
In the example given, the algebraic solution to the system of equations is the set of values for \( x \), \( y \), and \( z \) that satisfy all the equations at the same time. This solution is crucial because it represents the point where all equations in the system intersect, symbolizing the common solution.
The process of finding an algebraic solution often requires a combination of arithmetic operations and logical thinking. It is notably satisfying when we arrive at the concrete values of \( x = 6 \), \( y = 3 \), and \( z = 5 \) that solve the system, providing a clear set of results to a potentially complex problem.
Other exercises in this chapter
Problem 40
Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{c} {4 x-5 y \geq-20} \\ {x \geq-3} \
View solution Problem 40
write the partial fraction decomposition of each rational expression. $$\frac{3 x^{3}-6 x^{2}+7 x-2}{\left(x^{2}-2 x+2\right)^{2}}$$
View solution Problem 41
Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution s
View solution Problem 41
Solve each system by the method of your choice. $$ \left\\{\begin{array}{l} {x^{2}+y^{2}+3 y=22} \\ {2 x+y=-1} \end{array}\right. $$
View solution