Solving Equations Using Graphs

2x-6=0 means y=0. Where on the grid is y=0?

Find where the line is at a height of 6, then read the value of x there.

The solution is where the two lines cross.

x^2-x-6=0 means y=0. There are two solutions.

Where on a graph is y=0?

Look at where the curve meets the x-axis. Does it cross it, or just touch it?

1. x^2-6x+9=0 where y=0, which is on the x-axis.
2. The curve does not cross the x-axis. Its lowest point just touches the x-axis at x=3.
3. So there is one solution: x=3. Check: 3^2-6\times3+9=9-18+9=0.

Find the points where the curve and the line y = 3 meet, then read the x-coordinates.

1. The solutions are where the curve meets the line y=3.
2. The curve meets the line at x=-4 and at x=2.
3. So x=-4 or x=2. Check: (-4)^2+2\times(-4)-5=16-8-5=3 and 2^2+2\times2-5=4+4-5=3.

The curve shows y = x^2 - 4x. Which value of y does the equation ask for?

Part (a)

1. x^2-4x is the y value of the curve.
2. So x^2-4x=5 where y=5. Ravi should draw the line y=5.

Part (b)

1. The line y=5 meets the curve at x=-1 and at x=5.
2. So x=-1 or x=5. Check: (-1)^2-4\times(-1)=1+4=5 and 5^2-4\times5=25-20=5.

Find the point where the two lines cross. Take care with the negative numbers on each axis.

Part (a)

1. A point on a line satisfies its equation. The crossing is on both lines, so its x and y make both equations true at once.
2. Find the crossing, then trace vertically to the x-axis. This gives x=-2.

3. The lines meet at (-2,-4), so x=-2.

Part (b)

1. The crossing lies on both lines, so it gives the same x and y for both equations. From the crossing (-2,-4), trace horizontally to the y-axis: y=-4.
2. Check: -2-2=-4 and -2\times(-2)-8=4-8=-4.

The equation asks for a value of x. Which coordinate of the crossing point is that?

1. Where the lines cross, x+2 and 8-x have the same value.
2. That happens at x=3. The 5 is the y-coordinate: the value of both sides.
3. Check: 3+2=5 and 8-3=5. The solution is x=3.

Each small square across is 0.2. Find where the curve crosses the x-axis.

1. The solutions are where the curve crosses the x-axis (y=0).
2. Each small square across is 0.2.
3. The curve crosses the x-axis at about x=-1.3 and at about x=2.3.

Draw the line y = 3 across the grid. Where does it meet the curve?

1. Find 3 on the y-axis and go across in both directions to the curve. (You could draw the line y=3.)
2. The curve is at a height of 3 when x=-3 and when x=2.
3. So x=-3 or x=2. Check: (-3)^2+(-3)-3=9-3-3=3 and 2^2+2-3=3.

Rearrange x^2 - 2x = 8 so that one side is x^2 - 2x - 8. What must the other side be?

1. Subtract 8 from both sides: x^2-2x=8 is the same as x^2-2x-8=0.
2. So find where the curve crosses the x-axis (y=0): at x=-2 and at x=4.
3. So x=-2 or x=4. Check: (-2)^2-2\times(-2)=4+4=8 and 4^2-2\times4=16-8=8.

Check the scale on each axis before you read the crossing point. They are not the same.

1. The two axes have different scales: each small square across is 1, but each small square up is 5.
2. The lines cross at x=3, y=35.
3. Check: 10\times3+5=35 and 50-5\times3=35. The point is (3, 35).

For (b), each small square across is 0.5. Start at -2 on the y-axis.

Part (a)

1. y=2\times4-3=8-3=5.
2. The line goes through (4, 5), which agrees with the graph.

Part (b)

1. Find -2 on the y-axis and go across to the line, then up to the x-axis.
2. Each small square across is 0.5. The line is at y=-2 when x=0.5.
3. Check: 2\times0.5-3=1-3=-2. So x=0.5.

For (b), imagine the line y = 4 drawn across the grid. Each small square across is 0.2.

Part (a)

1. The turning point is the lowest point of the curve.
2. It is at (2, -3).

Part (b)

1. Draw the line y=4 across the grid.
2. Each small square across is 0.2. The line meets the curve at about x=-0.6 and at about x=4.6.
3. So x\approx-0.6 or x\approx4.6. Check: (-0.6)^2-4\times(-0.6)+1=0.36+2.4+1=3.76, close to 4.

The equations are not written as y = \ldots, but the method is the same: find where the lines cross.

Part (a)

1. A point on a line satisfies its equation. The crossing is on both lines, so its x and y make both equations true at once.
2. Find the crossing, then trace vertically to the x-axis. This gives x=4.

3. The lines meet at (4,-2), so x=4.

Part (b)

1. The crossing lies on both lines, so it gives the same x and y for both equations. From the crossing (4,-2), trace horizontally to the y-axis: y=-2.
2. Check: 3\times4+(-2)=12-2=10 and 3\times(-2)=-6, 4-10=-6.

Picture the horizontal line y = k moving up and down. Look at the lowest point of the curve.

Part (a)

1. Imagine the line y=5 across the grid.
2. It meets the curve twice, so there are 2 solutions.

Part (b)

1. The lowest point of the curve is at y=-3.
2. The line y=-4 is below it, so it never meets the curve: 0 solutions.

Part (c)

1. A horizontal line meets the curve exactly once only at the turning point, the lowest point of the curve.
2. The turning point is (-2, -3), so the line is y=-3 and k=-3.

Each small square across is 0.2 seconds and each small square up is 2.5 metres.

Part (a)

1. The greatest height is at the top of the curve.
2. It is 20 metres.

Part (b)

1. Go across from 15 on the height axis. It meets the curve twice.
2. Go down from each point to the time axis.
3. The ball is 15 m high after 1 second (going up) and after 3 seconds (coming down).

Part (c)

1. The ball is higher than 15 m between the two times in (b).
2. 3-1=2 seconds.

For (c), first find when candle A's height reaches 0.

Part (a)

1. The candles are the same height where the two lines cross.
2. The lines cross at 4 hours.

Part (b)

1. At the crossing point the height is 10 cm.

Part (c)

1. Candle A has burnt down when its height is 0: at 8 hours.
2. Go up from 8 hours to the line for candle B: 5 cm.
3. Candle B is 5 cm tall.