Let the two numbers be $x$ (smaller) and $y$ (larger). We can write the problem as a system of two equations:

1. $x + y = 42$
2. $x + 4y = 147$

Step 1: Solve for $x$ in terms of $y$ from the first equation

From $x + y = 42$,
$x = 42 - y$

Step 2: Substitute $x = 42 - y$ into the second equation

Substitute into $x + 4y = 147$:
$(42 - y) + 4y = 147$ Simplify:
$42 + 3y = 147$ $3y = 105$ $y = 35$

Step 3: Solve for $x$

Substitute $y = 35$ into $x = 42 - y$:
$x = 42 - 35$ $x = 7$

Final Answer:

The two numbers are:
Smaller number: $x = 7$
Larger number: $y = 35$

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Would you like further clarification or details on any step?

Related Questions:

1. How do you solve systems of linear equations using substitution?
2. Can the problem be solved using the elimination method instead?
3. What are other word problems involving systems of equations?
4. How can this problem be visualized graphically?
5. How can we verify the solution?

Tip:

Always check your solution by substituting the values back into the original equations to ensure they satisfy both equations.