To find the number of boys and girls who took part in the Mathematics quiz, let's follow a step-by-step approach.
1. Define Variables:
- Let the number of boys be [tex]\( b \)[/tex].
- Let the number of girls be [tex]\( g \)[/tex].
2. Form the Equation:
- We know that the total number of students is 10. Therefore, we can write the equation:
[tex]\[
b + g = 10 \quad \text{(Equation 1)}
\][/tex]
- We are also given that the number of girls is 4 more than the number of boys. This can be written as:
[tex]\[
g = b + 4 \quad \text{(Equation 2)}
\][/tex]
3. Substitute Equation 2 into Equation 1:
- Substitute [tex]\( g \)[/tex] from Equation 2 into Equation 1:
[tex]\[
b + (b + 4) = 10
\][/tex]
- Simplify the equation:
[tex]\[
2b + 4 = 10
\][/tex]
- Subtract 4 from both sides:
[tex]\[
2b = 6
\][/tex]
- Divide by 2:
[tex]\[
b = 3
\][/tex]
4. Find the Number of Girls:
- Substitute [tex]\( b = 3 \)[/tex] back into Equation 2:
[tex]\[
g = b + 4
\][/tex]
[tex]\[
g = 3 + 4
\][/tex]
[tex]\[
g = 7
\][/tex]
Therefore, the number of boys who took part in the quiz is 3, and the number of girls is 7.
### Graphical Solution:
To visualize the solutions graphically, we can plot the two equations on a graph:
1. Plot Equation 1 ([tex]\( b + g = 10 \)[/tex]):
- The intercepts are:
- If [tex]\( b = 0 \)[/tex], [tex]\( g = 10 \)[/tex]
- If [tex]\( g = 0 \)[/tex], [tex]\( b = 10 \)[/tex]
- Plot these points (0, 10) and (10, 0) and draw the line connecting them.
2. Plot Equation 2 ([tex]\( g = b + 4 \)[/tex]):
- The intercepts are:
- Choose some values for [tex]\( b \)[/tex] and solve for [tex]\( g \)[/tex]:
- If [tex]\( b = 0 \)[/tex], [tex]\( g = 4 \)[/tex]
- If [tex]\( b = 2 \)[/tex], [tex]\( g = 6 \)[/tex]
- Plot points (0, 4) and (2, 6) and draw the line connecting them.
3. Find the Intersection:
- The intersection point of the two lines gives the solution to the system of equations.
- From the graph, you will see that the lines intersect at the point (3, 7).
This confirms our calculated solution: there are 3 boys and 7 girls who took part in the quiz.
