Question

2x+3 =3x+4
On graph two straight lines crossing at (-1,1)

Answer 1

To solve the equation \(2x + 3 = 3x + 4\) and verify that the lines cross at the point (-1, 1), we will first solve for \(x\).

1. Start with the original equation:
\[
2x + 3 = 3x + 4
\]

2. Rearrange the equation to isolate \(x\):
\[
2x + 3 - 3 = 3x + 4 - 3
\]
\[
2x + 3 - 4 = 3x
\]
\[
2x - 3x = 4 - 3
\]
\[
-x = 1
\]

3. Multiply both sides by -1:
\[
x = -1
\]

Now, we'll find the corresponding \(y\) value when \(x = -1\) for both lines represented by the original equation.

### Find \(y\) for the first line:
Using \(y = 2x + 3\):
\[
y = 2(-1) + 3 = -2 + 3 = 1
\]
So the point on the first line is \((-1, 1)\).

### Find \(y\) for the second line:
Using \(y = 3x + 4\):
\[
y = 3(-1) + 4 = -3 + 4 = 1
\]
So the point on the second line is also \((-1, 1)\).

### Conclusion:
Both lines meet at the point \((-1, 1)\), confirming that they intersect there.

### Graphing the Lines:
1. **First line**: \(y = 2x + 3\)
- When \(x = 0\), \(y = 3\) (y-intercept)
- When \(x = -1\), \(y = 1\)
- When \(x = 1\), \(y = 5\)

2. **Second line**: \(y = 3x + 4\)
- When \(x = 0\), \(y = 4\) (y-intercept)
- When \(x = -1\), \(y = 1\)
- When \(x = 1\), \(y = 7\)

### Summary:
The two lines represented by the equations \(y = 2x + 3\) and \(y = 3x + 4\) intersect at the point \((-1, 1)\).