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Q7:
Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
Solution :
Given: Two linear equations in two variables:
1) $x - y + 1 = 0$
2) $3x + 2y - 12 = 0$
To Find: The coordinates of the vertices of the triangle formed by these two lines and the x-axis, and to represent the region graphically.
Step 1: Finding coordinates for the first equation $x - y + 1 = 0$
Rearranging the equation: $y = x + 1$.
If $x = -1$, then $y = 0$. Point: $(-1, 0)$
If $x = 0$, then $y = 1$. Point: $(0, 1)$
If $x = 1$, then $y = 2$. Point: $(1, 2)$
Step 2: Finding coordinates for the second equation $3x + 2y - 12 = 0$
Rearranging the equation: $2y = 12 - 3x \implies y = \frac{12 - 3x}{2}$.
If $x = 0$, then $y = 6$. Point: $(0, 6)$
If $x = 2$, then $y = 3$. Point: $(2, 3)$
If $x = 4$, then $y = 0$. Point: $(4, 0)$
Step 3: Determining the intersection point of the two lines
We solve the system of equations:
(i) $x - y = -1 \implies y = x + 1$
(ii) $3x + 2y = 12$
Substitute (i) into (ii):
$3x + 2(x + 1) = 12$
$3x + 2x + 2 = 12$
$5x = 10 \implies x = 2$
Substitute $x = 2$ into $y = x + 1$:
$y = 2 + 1 = 3$
The intersection point is $(2, 3)$.
Step 4: Identifying the vertices of the triangle
The triangle is formed by the two lines and the x-axis ($y=0$).
- The first line $x - y + 1 = 0$ intersects the x-axis at $y=0 \implies x = -1$. Vertex: $(-1, 0)$.
- The second line $3x + 2y - 12 = 0$ intersects the x-axis at $y=0 \implies 3x = 12 \implies x = 4$. Vertex: $(4, 0)$.
- The two lines intersect at $(2, 3)$. Vertex: $(2, 3)$.
Final Answer: The vertices of the triangle formed are $(-1, 0)$, $(4, 0)$, and $(2, 3)$.
More Questions from Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.1
- Q1(i): Form the pair of linear equations in the following problems, and find their solutions graphically. (i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
- Q1(ii): Form the pair of linear equations in the following problems, and find their solutions graphically. (ii) 5 pencils and 7 pens together cost ` 50, whereas 7 pencils and 5 pens together cost ` 46. Find the cost of one pencil and that of one pen.
- Q2(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0; 7x + 6y – 9 = 0
- Q2(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
- Q2(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (iii) 6x – 3y + 10 = 0; 2x – y + 9 = 0
- Q3(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7
- Q3(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (ii) 2x – 3y = 8 ; 4x – 6y = 9
- Q3(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iii) $\frac{3}{2}x + \frac{5}{3}y = 7$ ; 9x – 10y = 14
- Q3(iv): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iv) 5x – 3y = 11 ; – 10x + 6y = –22
- Q3(v): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (v) $\frac{4}{3}x + 2y = 8$ ; 2x + 3y = 12
- Q4(i): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) x + y = 5, 2x + 2y = 10
- Q4(ii): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (ii) x – y = 8, 3x – 3y = 16
- Q4(iii): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
- Q4(iv): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
- Q5: Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
- Q6(i): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines
- Q6(ii): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (ii) parallel lines
- Q6(iii): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (iii) coincident lines
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