Problem 85 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the system. Then check your solution. $$ \begin{aligned} &2 a+3 b=17\\\ &3 a+4 b=24 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(a=4\) and \(b=3\).

Step 1: Set up the system of equations

The system of equations is already set up: \[\begin{aligned}&2a+3b=17\&3a+4b=24\end{aligned}\]

Step 2: Solve the system using linear combinations

One way to solve this system is to multiply each equation by a constant to make the coefficient of \(b\) in the first equation equal and opposite to the coefficient of \(b\) in the second equation. Multiplying the first equation by 4 and the second equation by 3 we obtain: \[\begin{aligned}&8a+12b=68\&9a+12b=72\end{aligned}\]Now, if we subtract the first line from the second line we get:\[\begin{aligned}&a=4\end{aligned}\]

Step 3: Substitute \(a\) into the original equation

Substitute \(a=4\) into the first original equation, we get \[\begin{aligned}&2(4)+3b=17\&8+3b=17\&3b=9\&b=3\end{aligned}\]

Step 4: Checking the solution

Substitute \(a=4\) and \(b=3\) into the original second equation: \[3(4) + 4(3) = 12 + 12 = 24\], which is true. Therefore, the solution is correct.

Key Concepts

Linear Combinations

Linear combinations are a method used to solve systems of linear equations. In this approach, equations are multiplied by constants to align their coefficients. This alignment allows for the addition or subtraction of equations to eliminate one of the variables. In our example, the objective is to find values for the variables that satisfy both equations simultaneously. By multiplying each equation by a specific constant, we make the coefficients of one variable the same but opposite in sign, for example, the variable \( b \) in the equations becomes the focus:

• Multiply the first equation by 4: \( 8a + 12b = 68 \)
• Multiply the second equation by 3: \( 9a + 12b = 72 \)

We now have a set of equations where the \( b \) terms can cancel each other out when we subtract one from the other.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In mathematics, they are used to find the values of the unknowns that satisfy all of the equations in the system. An example system could be made of equations that form lines when graphed. Where these lines intersect represents the solution to the system.In the example:

• Equation 1: \( 2a + 3b = 17 \)
• Equation 2: \( 3a + 4b = 24 \)

To solve a system such as this, the linear combination method is highly effective. This involves adjusting the equations such that one variable can be easily eliminated, simplifying the process of finding the other variable.

Algebraic Solution

Solving equations algebraically involves using standard algebraic operations to simplify and find the unknown variable values. This is in contrast to graphical solutions, where the solution is found visually.In our worked solution, after we perform the linear combinations:

• We simplified: \( 9a + 12b - (8a + 12b) = 72 - 68 \)
• This simplified equation resolves to: \( a = 4 \)

With \( a \) known, we substitute it back into one of the original equations to find \( b \):

• Substitution into \( 2a + 3b = 17 \) gives us \( 2(4) + 3b = 17 \), simplifying to \( 3b = 9 \), so \( b = 3 \).

Finally, we check this solution by plugging \( a = 4 \) and \( b = 3 \) into the other equation to ensure both original equations are satisfied. This step confirms the accuracy of our algebraic solution, completing the procedure.