Answer:
- For equation: y = 2x + 3 , 2x - y = 5 There is no solution to this system of equations.
- For equation: y = 2x + 3 , 2x - y = 5 There is one solution to the system of equations.
- For equation: y = x + 4 , y = x + 4 There are infinite solutions to the system.
Step-by-step explanation:
- The equation of straight line is written as y = m x + c where 'm' is <u>slope</u>.
- y - intercept of line is value which intersect the point on y-axis.
- To find y -intercept , put x = 0 in equation.
- If slope of two lines are equal then lines are parallel.
- If the lines are not parallel, they will always intersect
<u>For equation: </u><u>y = 2x + 3 </u><u>,</u><u> 2x - y = 5</u>
Compare equation y = 2x + 3 with y = m x + c then we get <u>slope '</u><u>m</u><u>' is </u><u>2</u>.
Now, put x = 0 in y = 2x + 3 to get y-intercept
y = 2(0)+ 3
y = 0 + 3
y = 3
so, the<u> y-intercept is 3</u> .
Re-write this equation 2x - y = 5 in the slope intercept form;
Subtract 2x from both the sides of 2x - y = 5
2x - y - 2x = 5 - 2x
- y = - 2x + 5
Multiply both the sides by '-1'
y = 2x - 5
so, when we compare above equation with y = m x + c then we get slope 'm' is 2 .
Now, put x = 0 in y = 2x - 5 to get y-intercept
y = 2(0) - 5
y = 0 - 5
y = - 5
so, the<u> y-intercept is - 5</u> .
Equation y = 2x + 3 and 2x - y = 5 are <u>parallel</u> (since there slope are equal 'm = 2').
<u>There is no solution to this system of equations.</u>
<u>For equation: </u><u>3x - y = 5 </u><u>,</u><u> 2x + y = -3</u>
Re-write this equation 3x - y = 5 in the slope intercept form;
Subtract 3x from both the sides of 3x - y = 5
3x - y - 3x = 5 - 3x
- y = - 3x + 5
Multiply both the sides by '-1'
y = 3x - 5
so, when we compare above equation with y = m x + c then we get slope 'm' is 3 .
Now, put x = 0 in y = 3x - 5 to get y-intercept
y = 3(0) - 5
y = 0 - 5
y = - 5
so, the<u> y-intercept is - 5</u> .
Re-write this equation 2x + y = -3 in the slope intercept form;
Subtract 2x from both the sides of 2x + y = -3
2x - y - 2x = -3 - 2x
- y = - 2x - 3
Multiply both the sides by '-1'
y = 2x + 3
so, when we compare above equation with y = m x + c then we get slope 'm' is 2 .
Now, put x = 0 in y = 2x + 3 to get y-intercept
y = 2(0) + 3
y = 0 + 3
y = 3
so, the<u> y-intercept is 3</u> .
Equation 3x - y = 5 and 2x + y = -3 are <u>intersecting lines</u> (since their slope are not equal).
<u>There is one solution to the system of equations</u> (since the lines are intersect).
<u>For equation: </u><u>y = x + 4 </u><u>,</u><u> y = x + 4</u>
Compare equation y = x + 4 with y = m x + c then we get <u>slope '</u><u>m</u><u>' is 1</u>.
Now, put x = 0 in y = x + 4 to get y-intercept
y = 0+ 4
y = 4
so, the<u> y-intercept is 4</u> .
since equation of line are same therefore slopes are also equal
Equation y = x + 4<u> </u>and y = x + 4 are <u>parallel lines.</u>
<u>There are infinite solutions to the system. </u>( Since equivalent equation)
