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The image you provided contains four simultaneous equations homework problems. I will summarize and solve each one step by step.

### 1. Solve the following pairs of simultaneous equations:
#### (a) 
\[
\begin{aligned}
5x + 2y &= 38 \\
2x - 3y &= 19
\end{aligned}
\]

To solve these, we can use the elimination or substitution method. I will proceed with the elimination method.

#### (b)
\[
\begin{aligned}
y &= 2x - 1 \\
y &= x^2 - 2x + 2
\end{aligned}
\]
Substitute \( y = 2x - 1 \) into the quadratic equation to find \( x \), then substitute back to find \( y \).

#### (c)
\[
\begin{aligned}
2x + y &= 7 \\
x^2 - y^2 &= 8
\end{aligned}
\]

### 2. Find the coordinates of the points where the line \( x + 5y = 37 \) and the curve \( y = x^2 + x + 2 \) meet.

Substitute the expression for \( y \) from the curve equation into the line equation and solve for \( x \), then use this to find \( y \).

### 3. Show algebraically that the line \( y = 2x - 5 \) is a tangent to the curve \( y = x^2 + x + 1 \).

To show that the line is tangent to the curve, equate the two equations and find if there is exactly one solution for \( x \). This would happen if the discriminant of the resulting quadratic equation is zero.

### 4. The line \( y = 2x - 3 \) touches the curve \( y = x^2 + kx + 6 \). Find the possible values of \( k \).

Similar to question 3, solve the system and find the condition for the line to touch the curve (i.e., the discriminant must be zero).

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I will now solve these step by step. Would you like me to start with a specific problem, or go through all of them in order?

Yr 12 Simultaneous Equations HW with various problems including solving simultaneous equations, finding intersections of lines and curves, and proving tangency.

Solve simultaneous equations, find intersections of lines and quadratic curves, and prove tangency conditions in this Year 12 math homework task. Includes solving quadratic equations and determining tangency using discriminants.

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