91Ó°ÊÓ

Linear equations are algebraic expressions where the highest power of the variable is one. They have the general form \[ ax + by = c \], where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

Linear equations can represent real-world problems, often related to uniform motion or constant rates. In this problem, we have two linear equations:\[ 7a + 4b = 40 \] and \[ a + 3b = 30 \]. Simplifying these equations by combining like terms is crucial for solving them.

Such equations can be solved using several methods, including graphical representation, the substitution method, or the variable elimination method. Being able to identify and rearrange these linear equations correctly is key to finding the solution.

The substitution method involves solving one equation for one variable and substituting that solution into another equation. This is a powerful method to solve systems of linear equations.

In our example:\[ a + 3b = 30 \] was simplified to find \(a\):\[ a = 30 - 3b \].

By substituting \(a = 30 - 3b\) into the first simplified equation \[ 7a + 4b = 40 \], we get:\[ 7(30 - 3b) + 4b = 40 \]. After expanding and simplifying, we solve for \(b\):
\[ 210 - 21b + 4b = 40 \]\[ 210 - 17b = 40 \]\[ -17b = -170 \]\[ b = 10 \].

Once \(b\) is found, we substitute back into the simplified equation to find \(a\):\[ a = 30 - 3(10) \]\[ a = 0 \].

The substitution method is efficient and minimizes errors by reducing the number of steps in solving the system.

Variable elimination, also known as the elimination method, involves combining equations to cancel out one variable, making it easier to solve for the other.

Although not directly used in this particular exercise, understanding this method enhances comprehension. Here’s a quick overview:

• Add or subtract one equation from another to eliminate one of the variables.
• Solve the simplified equation for the remaining variable.
• Substitute the found value back into one of the original equations to find the other variable.

For example, if we had the equations:\[ 3x + 4y = 20 \] and\[ 5x + 4y = 30 \], we could subtract the first from the second:\[ (5x + 4y) - (3x + 4y) = 30 - 20 \]\[ 2x = 10 \]\[ x = 5 \].

Substitute \(x = 5\) back into one of the original equations to find \(y\). Thus, variable elimination helps in quickly solving systems of equations by breaking them down into simpler steps.