Question

solution:

1. y = 2x + 4 2y = 2x + 4

(grid graph with x from -10 to 10, y from -10 to 10)
solution:

Explanation:

Step1: Simplify the second equation

The second equation is \(2y = 2x + 4\). Divide both sides by 2 to simplify it: \(y = x + 2\).

Step2: Set the two equations equal

We have the first equation \(y = 2x + 4\) and the simplified second equation \(y = x + 2\). Set them equal to each other: \(2x + 4 = x + 2\).

Step3: Solve for x

Subtract \(x\) from both sides: \(2x - x + 4 = x - x + 2\), which simplifies to \(x + 4 = 2\). Then subtract 4 from both sides: \(x + 4 - 4 = 2 - 4\), so \(x = -2\).

Step4: Solve for y

Substitute \(x = -2\) into the second simplified equation \(y = x + 2\): \(y = -2 + 2 = 0\). Wait, let's check with the first equation: \(y = 2(-2) + 4 = -4 + 4 = 0\). Wait, that seems off. Wait, no, let's do it again. Wait, first equation: \(y = 2x + 4\), second equation: \(2y = 2x + 4\) or \(y = x + 2\). So set \(2x + 4 = x + 2\). Subtract \(x\) and 4: \(2x - x = 2 - 4\), so \(x = -2\). Then \(y = 2(-2) + 4 = 0\). Wait, but let's check the second equation: \(2y = 2x + 4\), plug in \(x = -2\), \(y = 0\): \(2(0) = 2(-2) + 4\) → \(0 = -4 + 4\) → \(0 = 0\), which is correct. Wait, but maybe I made a mistake. Wait, no, let's graph them mentally. First line: \(y = 2x + 4\) has y-intercept 4, slope 2. Second line: \(y = x + 2\) has y-intercept 2, slope 1. The intersection should be where \(2x + 4 = x + 2\), so \(x = -2\), \(y = 0\). Wait, but let's check again. Wait, maybe the problem is to find the intersection point. So the solution is the point \((-2, 0)\)? Wait, no, wait, let's do the algebra again. Wait, first equation: \(y = 2x + 4\), second equation: \(2y = 2x + 4\) → \(y = x + 2\). So set \(2x + 4 = x + 2\). Subtract \(x\): \(x + 4 = 2\). Subtract 4: \(x = -2\). Then \(y = 2(-2) + 4 = 0\). Yes, that's correct. So the solution is \(x = -2\), \(y = 0\), so the point \((-2, 0)\).

Wait, but maybe I messed up. Wait, let's check with the two original equations. First equation: \(y = 2x + 4\). If \(x = -2\), \(y = 0\). Second equation: \(2y = 2x + 4\). Plug in \(x = -2\), \(y = 0\): \(2(0) = 2(-2) + 4\) → \(0 = -4 + 4\) → \(0 = 0\), which is true. So the solution is \((-2, 0)\).

Wait, but maybe the problem is to solve the system of equations \(y = 2x + 4\) and \(2y = 2x + 4\). So the solution is the point where they intersect, which is \((-2, 0)\).

Answer:

The solution to the system of equations is \(x = -2\), \(y = 0\) or the point \((-2, 0)\).