Exercises 57–59 will help you prepare for the material covered in the next section. Solve: A + B = 3, 2A - 2B + C = 17, 4A - 2C =14

Verified step by step guidance

1. Step 1: Start by examining the system of equations: $A+B=3$, $2A-2B+C=17$, and $4A-2C=14$.
2. Step 2: Solve the first equation for one of the variables, for example, solve for $B$: $B=3-A$.
3. Step 3: Substitute $B=3-A$ into the second equation $2A-2B+C=17$ to eliminate $B$.
4. Step 4: Simplify the resulting equation from Step 3 to express $C$ in terms of $A$.
5. Step 5: Substitute the expression for $C$ from Step 4 into the third equation $4A-2C=14$ and solve for $A$.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this case, we have three equations involving the variables A, B, and C, which can be solved using methods such as substitution, elimination, or matrix operations.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equations. This reduces the number of equations and variables, making it easier to find the solution. For example, if we solve the first equation for A, we can substitute that value into the other equations to find B and C.

Linear Combination

Linear combination refers to the process of adding or subtracting equations to eliminate a variable, making it easier to solve the system. This technique is particularly useful when dealing with multiple equations, as it allows for simplification and can lead to a quicker solution. In the given problem, we can manipulate the equations to isolate variables and find their values.