Question

Use substitution to solve the linear system. $$\begin{aligned} &x+y=1\\\ &2 x-y=2 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 1\) and \(y = 0\)

Step-by-step solution

Preparation step: Rearrange first equation

Rearrange first equation to isolate one of the variables. An easy start would be to isolate x in the first equation, leading to: \(x = 1 - y\)

Substitute x in the second equation

Substitute the value of x (in terms of y) from the first equation into the second equation: \(2(1 - y) - y = 2\)

Solve for y in the substituted equation

Simplify and solve for y in the equation. On simplification, \(2(1 - y) - y = 2\) would give, \(2 - 2y - y = 2\). Further simplifying, we get \(-3y = 0\), so \(y = 0\)

Substitute y = 0 in the first equation

Substitute y = 0 into the rearranged first equation \(x = 1 - y\), we get \(x = 1 - 0 = 1\).

Verify the solution

To verify the solution, substitute x = 1 , y = 0 in the original two equations: The first equation becomes \(1 + 0 = 1\) and the second equation becomes \(2(1) - 0 = 2\), Both are true statements, so the solution is verified

Key concepts

Substitution Method

The substitution method is a powerful algebraic technique used to find the exact solution to a system of linear equations. It involves substituting one variable from one equation into the other equation. This method is particularly useful when equations are already solved for one variable, making it easy to replace that variable in another equation.

In our given exercise, once we solved the first equation for x, we effectively 'substituted' x as '1 - y' into the second equation. This helps to reduce the system of equations from two equations with two variables to a single equation with one variable, which is much simpler to solve. As a tip, always aim to isolate the variable that makes the substitution step easier and leads to less complicated algebraic operations.

Linear Equations

Linear equations are fundamental in algebra and represent straight lines on a graph. They have the general form of ax + by = c, where 'a', 'b', and 'c' are constants. A system of linear equations consists of two or more linear equations with the same variables. The goal in solving such a system is to find the point or points where the equations intersect, meaning the values of the variables that satisfy all equations simultaneously.

When we solved our system, we looked for the values of x and y that worked for both equations. These values, once found, give us the intersection point of the lines, should these equations be graphed. It's crucial to understand that the equations in a system must be independent to have a unique solution—otherwise, they may be parallel (no solution) or coincident (infinite solutions).

Algebraic Methods

Algebraic methods encompass various techniques to manipulate and solve equations, such as substitution, elimination, and using matrices. These methods are rooted in the properties of equality and the distributive, associative, and commutative properties of numbers.

For instance, when we simplified the second equation after substitution, we used the distributive property to expand 2(1 - y) and the properties of equality to rearrange and solve for y. It's essential to be systematic and careful with each algebraic step to avoid simple errors, which can lead to incorrect solutions. Patience and practice with these methods will lead to mastery and an enhanced ability to approach more complex algebraic problems.

System of Equations

A system of equations is a set that includes two or more equations, each with the same set of variables. The solutions to a system are the values for the variables that satisfy all equations within the system at the same time. Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (the same line).

Our exercise presented a system with a unique solution. By following the correct steps, we substituted, reduced, solved for one variable, and then the other. Finally, we verified our solution by plugging the values back into the original equations. Always remember to check your solutions to ensure they meet the requirements of all the original equations - this verification step is crucial to confirm your solution is correct.