Question

Solve each system of equations by addition-subtraction, or by substitution. Check some by graphing. $$\begin{aligned} &y=9-3 x\\\ &x=8-2 y \end{aligned}$$

Short answer

The solution to the system of equations is x = -2 and y = 15.

Step-by-step solution

Express y in terms of x

The first equation is already solved for y: y = 9 - 3x.

Substitute y into the second equation

Replace y with 9 - 3x in the second equation: x = 8 - 2(9 - 3x).

Simplify the second equation

Simplify the equation from Step 2 to solve for x: x = 8 - 18 + 6x -> 5x = -10 -> x = -2.

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.

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 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.

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.

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:

• $$y = 9 - 3x$$ (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)

Plotting these on a graph, the intersection represents the solution (-2, 15). This method provides a visual representation and can be especially useful for understanding the nature of solutions (one solution, no solution, or infinitely many solutions). However, it's less precise than algebraic methods for finding exact solution values.

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:

• 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.

This algebraic strategy is typically preferred for its accuracy and ability to provide definitive answers, regardless of the number of solutions the system may be subject to.