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?

Short 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-by-step solution

Rearrange the first equation

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

Substitute into the second equation

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

Solve for x

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

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, \(y = 3\).

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.