Problem 21
Question
Solve each system of equations by addition-subtraction, or by substitution. Check some by graphing. $$\begin{aligned} &y=9-3 x\\\ &x=8-2 y \end{aligned}$$
Step-by-Step Solution
Verified Answer
The solution to the system of equations is x = -2 and y = 15.
1Step 1: Express y in terms of x
The first equation is already solved for y: y = 9 - 3x.
2Step 2: Substitute y into the second equation
Replace y with 9 - 3x in the second equation: x = 8 - 2(9 - 3x).
3Step 3: Simplify the second equation
Simplify the equation from Step 2 to solve for x: x = 8 - 18 + 6x -> 5x = -10 -> x = -2.
4Step 4: Solve for y using the value of x
Plug x = -2 into the first equation to find y: y = 9 - 3(-2) -> y = 9 + 6 -> y = 15.
5Step 5: Check the solution by graphing (optional)
Graph both equations on the same coordinate system to check if the point (-2, 15) is the point of intersection.
Key Concepts
Addition-Subtraction MethodSubstitution MethodGraphical Solution of EquationsAlgebraic Methods
Addition-Subtraction Method
The addition-subtraction method, also known as the elimination method, is used to solve systems of linear equations. The goal is to eliminate one variable by adding or subtracting the equations from one another.
For example, consider the system of equations:
$$\begin{aligned} y &= 9-3x \ x &= 8-2y \ \text{Step 1: Arrange in standard form} \ a_1x + b_1y &= c_1 \ a_2x + b_2y &= c_2 \text{Step 2: Multiply to obtain opposites} \ a_1x + b_1y &= c_1 \ a_2x - b_1y &= c_2 \text{Step 3: Add or subtract equations} \ (a_1 + a_2)x &= c_1 + c_2 \text{ (assuming } b_1y \text{ is eliminated)} \text{Step 4: Solve for } x \ x &= \frac{c_1 + c_2}{a_1 + a_2} \text{Step 5: Substitute to find } y \text{Step 6: Validate solution}Using this approach, you sequentially eliminate variables to simplify the equations until you solve for the unknowns. However, it's important to check the solution in both original equations to ensure it works for both, not just one.
For example, consider the system of equations:
$$\begin{aligned} y &= 9-3x \ x &= 8-2y \ \text{Step 1: Arrange in standard form} \ a_1x + b_1y &= c_1 \ a_2x + b_2y &= c_2 \text{Step 2: Multiply to obtain opposites} \ a_1x + b_1y &= c_1 \ a_2x - b_1y &= c_2 \text{Step 3: Add or subtract equations} \ (a_1 + a_2)x &= c_1 + c_2 \text{ (assuming } b_1y \text{ is eliminated)} \text{Step 4: Solve for } x \ x &= \frac{c_1 + c_2}{a_1 + a_2} \text{Step 5: Substitute to find } y \text{Step 6: Validate solution}Using this approach, you sequentially eliminate variables to simplify the equations until you solve for the unknowns. However, it's important to check the solution in both original equations to ensure it works for both, not just one.
Substitution Method
The substitution method is best applied when one equation in the system is solved for one variable, which can then be substituted into the other equation. This method often simplifies the calculation because it reduces the problem to a single variable equation.
Let's elaborate using the provided exercise:
$$y = 9 - 3x$$You would substitute this expression for y into the second equation:
$$x = 8 - 2y$$
After substitution, you solve the equation for x, and once found, you plug the value back into any of the original equations to find the second unknown. This process typically involves:
1. Isolation of one variable in one equation.
2. Substitution of the expression into the other equation.
3. Solution of the new single-variable equation.
4. Back-substitution to find the other variable.
Let's elaborate using the provided exercise:
$$y = 9 - 3x$$You would substitute this expression for y into the second equation:
$$x = 8 - 2y$$
After substitution, you solve the equation for x, and once found, you plug the value back into any of the original equations to find the second unknown. This process typically involves:
1. Isolation of one variable in one equation.
2. Substitution of the expression into the other equation.
3. Solution of the new single-variable equation.
4. Back-substitution to find the other variable.
Graphical Solution of Equations
A graphical solution involves plotting each equation on a coordinate system and finding the point(s) at which the lines intersect. This point corresponds to the solution of the system.
In our exercise, you'd plot both lines:
In our exercise, you'd plot both lines:
- $$y = 9 - 3x$$ (a straight line with intercept (0,9) and slope -3)
- $$x = 8 - 2y$$ (with intercept (8,0) and slope 1/2 after rearranging to y form)
Algebraic Methods
Algebraic methods for solving systems of equations, such as substitution or elimination, rely on manipulation of the equations to find exact solutions. These methods are systematic and can handle a wide range of problems, from simple to complex systems.
In the context of the exercise provided, the substitution method was an efficient choice since one variable was already isolated. To elaborate:
In the context of the exercise provided, the substitution method was an efficient choice since one variable was already isolated. To elaborate:
- Isolate one variable in any of the equations if not already done.
- Substitute the expression for the isolated variable into the other equation and solve for the remaining unknown.
- Substitute the found value into any of the original equations to find the value of the other variable.
- Validate the solution by plugging back the values into both original equations.
Other exercises in this chapter
Problem 20
Solve for \(x\) and \(y\) in terms of the other literal quantities. $$\begin{aligned} &m x+3 n y=2\\\ &2 m x+n y=1 \end{aligned}$$
View solution Problem 21
When writing Kirchhoff's voltage law for a certain three-loop network, we get the set of equations $$\begin{aligned}3 I_{1}+2 I_{2}-4 I_{3} &=4 \\\I_{1}-3 I_{2}
View solution Problem 21
Solve for \(x\) and \(y\) in terms of the other literal quantities. $$\begin{aligned} &2 p x+3 q y=3\\\ &3 p x+2 q y=4 \end{aligned}$$
View solution Problem 22
Solve each system of equations by addition-subtraction, or by substitution. Check some by graphing. $$\begin{aligned} &y=2 x-3\\\ &x=19-3 y \end{aligned}$$
View solution