Problem 19 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. $$ \begin{array}{r} {2 a=8} \\ {a+b=2} \end{array} $$

Step-by-Step Solution

Answer

The solution for the system of equations is \(a = 4\) and \(b = -2\).

Step 1: Isolate a variable in one of the equations

From the first equation, isolate a by dividing both sides of the equation by 2. This gives \(a = \frac{8}{2} = 4\).

Step 2: Substitute the isolated variable into the other equation

Substitute \(a = 4\) into the second equation to solve for b. This gives \(4 + b = 2\). To solve for b, subtract 4 from both sides of the equation to isolate b, giving \( b = 2 - 4 = -2\).

Step 3: Check the solution

Insert the obtained values \(a = 4\) and \(b = -2\) back into the original equations to check if they satisfy both the equations. For the first equation, \(2*4 = 8\) and for the second equation, \(4 - 2 =2\). Since both these are true, the solution is correct.

Key Concepts

Solving Linear Systems

Solving linear systems is an essential skill in algebra that requires finding values for variables that satisfy all given equations simultaneously. The substitution method, showcased in the provided exercise, is a common technique used to solve systems of linear equations. It involves isolating one variable in one of the equations and then replacing, or substituting, this variable's value in the other equation(s). By methodically applying this technique, students can find the solution that works for all equations in the system, leading to the values that constitute the solution of the system.

Isolating Variables

Isolating variables is a key step in many algebraic techniques including the substitution method. It means rearranging an equation so that one variable stands alone on one side of the equality sign. For instance, if the exercise gives the equation 2a = 8, we isolate a by dividing both sides by 2, yielding a = 4. By isolating variables correctly, students form a bridge that allows them to substitute this expression into another equation and proceed towards solving the system.

Linear Equations

Linear equations are algebraic statements that involve constants and linear terms. They are represented in the form ax + by = c, where a, b, and c are constants, and x and y are variables. The graph of a linear equation is a straight line, and in a system of linear equations, the solution is the point or points where the lines intersect. The problem at hand includes two linear equations that together form such a system that we want to solve.

Algebraic Equations

Algebraic equations are equations that include numbers, variables, and various arithmetic operations. They range from simple linear equations to more complex ones involving powers and roots. The substitution method specifically addresses the need to solve linear algebraic equations, which are the cornerstone of understanding algebra. By mastering the solving of these, students build a strong foundation that supports their progress in more complicated areas of mathematics.

Equation Solving

Equation solving is the process of finding the values of variables that make an equation true. In our exercise, we found that setting a = 4 and b = -2 satisfies both equations 2a = 8 and a + b = 2. The final step often involves checking the solution by substituting the variables' values back into the original equations to verify their correctness. This approach ensures the accuracy of the solution and reinforces the student's understanding of the relationships between the equations in the system.