Problem 49 - Algebra 1

Question

Use the linear system below. $$\begin{array}{l} y=x+3 \\ y=2 x+3 \end{array}$$ Solve the linear system using substitution. What does the solution mean?

Step-by-Step Solution

Answer

The solution to the system of equations is (0, 3). This point represents the intersection of the two lines represented by the equations in the system.

Step 1: Rearrange the first equation

The first equation is already rearranged to: \(y = x + 3\).

Step 2: Substitute into the second equation

Substitute \(y\) from the first equation into the second equation. This results in the following equation: \(x + 3 = 2x + 3\).

Step 3: Solve for x

Solving the equation from Step 2, we get \(x = 0\).

Step 4: Find the y-coordinate

After finding x=0, substitute this into the first equation to solve for y. This results in the following equation: \(y = 0 + 3\). Hence, \(y = 3\).

Step 5: Interpret the solution

The point of intersection of the lines represented by the given equations is at (0,3). This means that for these values of x and y, both equations are made true. In other words, x=0 and y=3 is a valid solution to both original equations.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of equations. It involves expressing one variable in terms of another and substituting this expression into the second equation. This simplifies the equations, making it easier to find the solutions.\\Here's how it works step-by-step:\

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• First, solve one of the equations for one of the variables. In this exercise, the first equation is already solved for y: \( y = x + 3 \).
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• Substitute this expression (\( y = x + 3 \)) into the other equation. This substitution changes the second equation from \( y = 2x + 3 \) to \( x + 3 = 2x + 3 \).
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• Next, solve the resulting one-variable equation. Here, solving \( x + 3 = 2x + 3 \) simplifies to \( x = 0 \).
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\The substitution method is particularly useful when one of the equations is already simplified, allowing for straightforward substitutions.

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. In our exercise, the system is composed of the following two linear equations:\

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• \( y = x + 3 \)
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• \( y = 2x + 3 \)
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\Each equation represents a straight line on a coordinate plane. The solution to the system is the point where these lines intersect, meaning that this point satisfies all the equations simultaneously.

The intersection can occur in three ways:\

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• One unique solution where the lines intersect at one point, as in our exercise.
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• No solution, meaning the lines are parallel and never meet.
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• Infinite solutions, indicating the lines are identical and overlap entirely.
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\Understanding systems of equations is crucial, as they model various real-life situations where multiple conditions or constraints are applied.

Solutions of Linear Systems

Solutions of linear systems are the values of the variables that satisfy all equations in the system. For the linear system given in the exercise, the solution tells us the coordinates where the two lines represented by the equations meet. In this case, the solution is the ordered pair \((x, y) = (0, 3)\).\\This solution can be interpreted as follows:\

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• \(x = 0\) and \(y = 3\) are the values that make both equations true.
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• Graphically, this means the point (0, 3) lies on both lines plotted on a graph, denoting it's their intersection point.
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\To check the correctness of a solution, substitute the values back into the original equations. If both equations hold true, the solution is valid. Verifying solutions ensures accuracy and reinforces an understanding of solving techniques.